Download Ch 8 Probability Notes

8.1 Introduction to Probability
Math 12
Recall:
If an experiment has n equally likely outcomes of which r outcomes
are favourable to event A, then the probability of event A is:
P( A)
r
n
Example 1: A die is rolled. What is the probability of…
a) rolling a 6?
b) rolling an even number?
c) rolling a 5?
d) not rolling a 5?
Example 2: Suppose one card is drawn from a standard deck of 52 cards. What is the probability that it is…
a) a face card?
b) the queen of diamonds?
c) a black ace?
d) not a black ace?
Example 3: A die is rolled and a card is chosen from a standard deck of 52 cards. What is the probability of…
a) rolling a 4 and choosing a king?
b) rolling a 5 or 6 and not choosing a face card?
c) not rolling a prime number and choosing a black jack?
d) rolling a 1 and choosing the eight of clubs?
Using Tree Diagrams/Sample Spaces to Determine Probabilities
Draw a tree diagram and list the sample space to answer the following questions:
1. What is the probability that in a family of 3 children there are exactly 2 girls?
Assumption:
2. Consider the diagram below. When the pointer is spun twice, find the probability that…
a) the same number comes up on each spin.
b) the numbers have a sum of 8.
c) the sum of the numbers is odd.
3. Two marbles are randomly selected from the bag without replacement.
Find the probability that you choose two differently coloured marbles.
Assignment: EXERCISES 8.1: Probability Terminology and Notation
EXERCISES 8.1
Probability Terminology and Notation
1) A machine produces batteries. As part of a quality control process, six batteries are randomly selected
every hour and tested. The number of defective batteries is noted.
a) List the sample space.
b) List the following events as subsets of the sample space:
i)
At least two batteries are defective
ii)
At most two batteries are defective
c) List the complement of the following events as subsets of the sample space:
i)
Three batteries are defective
ii)
At most five batteries are defective
2) Eight coins are tossed and the number of tails is counted
a) List the sample space.
b) List the event “more tails than heads” in terms of the sample space.
c) List the event “more heads than tails” in terms of the sample space.
d) List the complement of the following events as subsets of the sample space:
i)
An even number of tails
ii)
At least one tail
3) A committee of two people is to be selected from: Alex (A), Bob (B), Carol (C), Dave (D), and Ed (E)
a) List the sample space
b) List the following events as subsets of the sample space:
i)
Alex is on the committee
ii)
Neither Carol nor Dave is on the committee
c) Assuming each element of the sample space to be equally likely, determine
i)
P(Alex is on the committee)
ii)
P(Bob and Dave are both on the committee)
iii)
P(Alex is not on the committee)
4) Packet 1 contains an orange candy, a lemon candy and a strawberry candy. Packet 2 contains an orange
candy, a strawberry candy and a raspberry candy. Without looking in the packets a student takes one
candy from packet 1, then one candy from packet 2.
a) Use a tree diagram to determine the sample space for the experiment.
b) List the event “both candies are the same type” as a subset of the sample space.
c) List the event “a strawberry candy is not chosen” as a subset of the sample space.
d) State the probability of the following events:
i)
Both candies are the same
ii)
A strawberry candy is not chosen
iii)
At least one orange candy is chosen
iv)
The candies are different types
e) Explain the relationship between the answers to parts i) and iv)
5) Four dimes are tossed by student 1 and five quarters are tossed by student 2. The number of heads is
recorded by each student.
a) List the sample space as an array of ordered pairs
b) List the event “student 1 recorded more heads than student 2” in terms of the sample space
c) State the probability of the following events:
i)
Student 1 recorded more heads than student 2
ii)
Student 1 recorded less heads than student 2
d) Is the event in c)ii) the complement of the event in c)i)? Explain
6) Rebecca, Elizabeth, and Jenny each toss a coin to see who is going to pay for coffee. If one of the
students throws a different outcome from the other two then that student buys all three coffee. If all the
student throw the same outcome then each student pays for their own coffee.
a) Use a tree diagram to construct a sample space for tossing three coins
b) State the probabilities of the following events:
i)
Rebecca buys all the coffees
ii)
Each student buys their own
iii)
Jenny gets a coffee free
7) An experiment consists of rolling a die, flipping a coin, and spinning a spinner divided into 5 regions.
How many elements are there in the sample space of this experiment?
8) If P( X ) 0.2 , then what is P( X ) ?
9) Box A contains a red marble, a blue marble and a yellow marble. Box B contains a red marble, a blue
marble, a yellow marble and a green marble. One marble is taken from each box. What is the
probability, to the nearest hundredth, that both marbles are the same colour?
Answer Key:
1) a) {0, 1, 2, 3, 4, 5, 6} b) i) {2, 3, 4, 5, 6}
ii) {0, 1, 2} c) i) {0, 1, 2, 4, 5, 6} ii) {6}
2) a) {0, 1, 2, 3, 4, 5, 6, 7, 8}
b) {5, 6, 7, 8} c) {0, 1, 2, 3} d) i) {1, 3, 5, 7} ii) {0}
3) a) {AB, AC, AD, AE, BC, BD, BE, CD, CE, DE}
3
2
1
b) i) {AB, AC, AD, AE} ii) {AB, AE, BE}
c) i)
ii)
iii)
5
5
10
4) a) {OO, OS, OR, LO, LS, LR, SO, SS, SR} b) {OO, SS}
d) i)
2
9
ii)
4
9
iii)
5) a) 30 pairs from (0, 0) to (4, 5)
c) i)
1
3
ii)
1
2
5
9
iv)
7
9
e) Event iv) is the complement of event i)
b) {(1, 0), (2, 0), (3, 0), (4, 0), (2, 1), (3, 1), (4, 1), (3, 2), (4, 2,), (4, 3)}
d) No, because the outcomes where the # of heads by each student is the same.
6) a) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} b) i)
7) 60
c) {OO, OR, LO, LR}
8) 0.8
9) 0.25
1
4
ii)
1
4
iii)
1
2
8.2 VENN DIAGRAMS
Math 12
Venn diagrams are illustrations used in the branch of mathematics known as set theory. They show all of the
possible mathematical or logical relationships between sets (groups of things).
A simple box can represent a specific sample space and an enclosed loop can represent a certain event. For
example, in a class of 32 students, 12 students like watching the Vancouver Canucks, and 15 students like
watching the BC Lions. Eight students like watching both.
Study the Venn Diagrams below:
1. Event C: like watching the Canucks
2. Event C (complement of event C): don’t like watching the Canucks
The event that C does not occur is the complement of the event C, and is denoted by C .
Note:
P(C) + P( C ) = 1
3. Event L: like watching the Lions
4. Event C and L: like watching the Canucks and like watching the Lions
5. Event C or L: like watching the Canucks or like watching the Lions
Example 1.
If each outcome in the sample space is equally likely to occur, find the following:
a. P(D) =
b. P(W) =
c. P(D and W) =
d. P(D or W) =
e. P( D ) =
f. P( D and W) =
g. P( D or W) =
Example 2. A black die and white die are rolled. Consider the following events:
Event E: The sum of the two die is at least 8
Event L: The black die is 1 less than the white die
WHITE DIE
a. P(E) =
1
b. P(L) =
2
3
4
5
6
1
d. P(E or L) =
e. P( E ) =
BLACK DIE
c. P(E and L) =
f. P( E and L) =
2
3
4
5
6
g. P( E or L) =
Example 3. Two dice are rolled. Use a sample space to determine the probability of each event.
a. The difference of the two dice is at least 4.
b. The sum of the two dice is less than 4.
c. The difference is at least 4 or the sum is less than 4.
d. The difference is at least 4 and the sum is less than 4.
Example 4. Two dice are rolled. Determine the probability of each event.
a. The sum is an even number.
b. The sum is either 6 or 12
c. A double is rolled.
d. The sum is at least 9.
Example 5.
In a survey, 42% of households contacted owned a home computer and 13% owned a home
entertainment system. Eight percent of all households contacted owned both. What is the
probability that a randomly selected household will own either? Include a Venn diagram.
Example 6.
A market study found that 50% of a neighbourhood like Japanese food while 60% like
Vietnamese food. Thirty percent like both. What is the probability that a randomly selected
resident will like Japanese food but not Vietnamese food? Include a Venn diagram.
Example 7.
A TV station determined that 30% of its teenage viewers watch sports programs and 60% watch
soaps. Twenty percent watch neither. What is the probability that randomly selected teenage
viewer watches both?
Math 12
8.3 The Event ‘A or B’
Examine the following cases. Each of the outcomes in the sample space is equally likely to occur.
Case #1:
Events A and B are MUTUALLY EXCLUSIVE.
Find the following probabilities by counting the outcomes.
P(A) =
P(B) =
A
B
P(A or B) =
P(A and B) =
What is the relationship between P(A or B), P(A), and P(B)?
Events A and B in case #1 are said to be mutually exclusive because they share no outcomes.
Case #2:
Events A and B are NOT MUTUALLY EXCLUSIVE.
Find the following probabilities by counting the outcomes.
P(A) =
P(B) =
A
B
P(A or B) =
P(A and B) =
What is the relationship between P(A or B), P(A), P(B), and P(A and B)?
Addition Law
General Case:
P(A or B) =
Special Sub-case: If A and B are mutually exclusive then
P(A or B) =
Example 1: One card is randomly drawn from a deck of 52 cards.
S: The card is a spade.
R: The card is red.
F: The card is a face card.
a) Which of these three events are mutually exclusive?
b) Determine the following probabilities:
P(S) =
P(R) =
P(S and R) =
P(S and F) =
P(F) =
c) Using the Addition Law, find the following:
P(S or R) =
P(S or F) =
Example 2: There are 30 students in a class. Sixteen students have a cell phone and 10 have an iPod. Of these
students, 6 have both. What is the probability that a randomly selected student in the class has a cell phone or an
iPod?
Solution a) Use the formula:
Solution b) Use a Venn diagram:
Example 3: Mary spends her whole weekend working on a Geography project and a book report. She thinks
she has a 70% chance of getting an „A‟ on the project, but just a 50% chance of getting an „A‟ on the book
report. She puts only a 25% chance of not getting an „A‟ on either. Find the probability that she gets…
a) at least one „A‟.
b) an „A‟ on both.
c) only one „A‟.
Assignment: EXERCISES 8.3: Mutually Exclusive Events and the Event “A or B”
Exercises 8.3
Mutually exclusive events and the event ‘A or B’
1) In each case, state whether the event A, B are mutually exclusive or not.
a) A card is drawn from a standard deck
Event A - a spade is selected
Event B – a club is selected
b) A card is drawn from a standard deck
Event A – a spade is selected
Event B – a black seven is selected
c) A card is drawn from a standard deck
Event A – a red card is selected
Event B – a face card is selected
d) Two dice are thrown
Event A – the dice both show odd #‟s
Event B – the total is 8
e) A chocolate bar is chosen
Event A – the bar contains raisins
Event B – the bar contains nuts
f) A student is chosen from this class
Event A – the student is left handed
Event B – the student is female
2) Use the Venn Diagram to state all pairs of the event which are mutually exclusive.
a)
b)
A
B
A
B
C
C
D
E
D
3) Meghan (M), Jolene (J), and Tara (T) are the only entrants in a competition for which there is a first and
second prize. Consider the experiment of selecting the two prize winners.
a) List the sample space
b) State the subsets of the sample space the events:
i) P – Meghan wins first prize
ii) Q – Meghan wins second prize
iii) R – Meghan wins a prize
iv) S – Jolene and Tara win the prizes
c) Which three of these events are mutually exclusive?
d) List all pairs of these events that are mutually exclusive.
4) Use the following information to determine whether the events A and B are mutually exclusive.
1
1
a) P( A)
, P( B)
, P( A and B) 0
2
5
b) P( A) 0.3, P( B) 0.2, P( A or B) 0.4
c) P( A)
2
, P( B)
5
1
, P( A or B)
4
13
20
5) Use the given information to calculate the probability of the indicated event.
a) Tanya and Fujiko are two of the competitors in the final of the long jump competition. The
1
2
probability that Tanya wins the competition is , and the probability that Fujiko wins is .
5
7
Calculate the probability that Tanya or Fujiko win the competition.
b) In a group of students, the probability that a student chosen at random walks to school is 0.35, and
the probability that a student has blond hair is 0.2. If the probability that a student walks to school or
has blond hair is 0.45, calculate the probability that a student walks to school and has blond hair.
c) Mohinder is playing with an incomplete desk of cards. When one card is selected at random from
the incomplete deck, the probability that it is a heart is 0.3, and the probability that it is a face card is
0.2. If the probability that it is a heart and a face card is 0.1, calculate the probability that it is a
heart or face card.
6) A card is drawn from a standard deck. Use an appropriate formula to determine that probability that:
a) A face card or club is drawn
b) A red card or club is drawn
c) An eight or a spade is drawn
d) A five of diamond or Queen is drawn
e) A black card or diamond is drawn
f) A jack or a red card is drawn
7) Allyson and Brittney were solving the following problem:
100 people in the community participated in a technology survey. It was found that 80 people have a
computer, 40 people have access to the internet, and 30 people have both a computer and access to the
internet. If one of these 100 people are chosen at random, what is the probability that the person has
a) A computer or access to the internet?
b) Neither?
c) Show how you got these answers using two methods (by Venn diagram, and by formula)
8) Mr. Spark has problems starting his two cars during the cold winter months. He decided to record the
number of times each car starts during a one month period in the winter. He attempted to start each car
every morning for thirty days. He recorded the following information:
- His first car started 20 times
- His second car started 18 times
- Both cars started 40% of the time
a) What is the probability that at least one of the cars start on any particular morning during the month?
b) What is the probability that he cannot start either of his two cars?
9) In a group of 80 elementary school children, 48 children liked cornflakes, 22 children liked both
cornflakes and oatmeal, and 20 children liked neither cornflakes nor oatmeal. Calculate the probability
that a child selected at random from the group liked only oatmeal.
10) In a school, 55% of the students have passed their English examination and 45% have passed their Math
examination. Comment critically on the following statement. “The probability the student chosen at
random has passed his or her English exam or math exam is 0.55 + 0.45 = 1”
11) In an election there are three candidates. Candidate A is twice as likely to win as candidate B, and
candidate B is twice as likely to win as candidate C.
a) What is the probability that candidate A wins the election?
b) What is the probability that candidate A or candidate B wins the election (to nearest hundredth)
Answer Key:
1) a) Yes
b) No
c) No
d) No
e) No
f) Prob. no (depends on class)
2) a) A and C, A and D
c) A and B, A and C, A and D, A and E, B and C, B and D, B and E, C and E
3) a) {MJ, MT, JM, JT, TM, TJ}
b) i) {MJ, MT} ii) {JM, TM}
iii) {MJ, MT, JM, TM}
iv) {JT, TJ}
c) P, Q, and S
d) P and Q, P and S, Q and S, R and S
4) a) Yes, since P (A and B) = 0
b) No, since P(A) + P(B)
P(A or B)
c) Yes, since P(A) + P(B) = P(A or B)
5) a)
17
35
b) 0.1
7) a)
9
10
b)
1
10
c) 0.4
6) a)
8) a)
13
15
11
26
b)
b)
2
15
3
4
c)
5
4
3
d)
e)
52
13
4
9)
3
20
10) The two events are not mutually exclusive, so the statement is false
11) a)
4
7
b) 0.86
f)
7
13
8.4 the Event ‘A and B’
Math 12
P( A and B) is the probability that event A occurs on a first and event B occurs on a second trial. To determine
the probability of event B, there must be a consideration on whether or not event A has occurred. Because of
this consideration, the symbol P( B | A) is used. This is termed the conditional probability of B given A.
NOTE: The P ( B |
A) may or may not be the same as P( B) , depending on whether the occurrence of event A affects the probability of event B.
Case #1:
Drawing Two Cards from a Deck, WITHOUT REPLACEMENT.
A card is drawn from a shuffled deck of 52 cards, and not replaced. Then a second card is drawn.
Let
F: The first card drawn is a spade
S: The second card drawn is a spade
The probability of drawing the first card is P ( F )
Remember, the first card is not replaced into the deck before the second card is drawn.
P(S given that the first card is spade) , that is, P( S | F )
Thus, P(S and F )
Event F and Event S are dependent events because the probability that Event S occurs depends on the
occurrence of Event F.
Case #2:
Drawing Two Cards from a Deck, WITH REPLACEMENT.
A card is drawn from a shuffled deck of 52 cards, then replaced. Then a second card is drawn.
Let
F: The first card drawn is a spade
S: The second card drawn is a spade
The probability of drawing the first card is P ( F )
Remember, in this situation, the first card is replaced before the second card is drawn.
P(S given that the first card is spade) , that is, P( S | F )
Thus, P(S and F )
In this case, P( S | F ) is the same as P( S ) . The Event F and Event S are independent events because the
probability that event S occurs does not depend on the occurrence of event F.
Multiplication Law
General Case:
P(A and B) = P(A) • P(B|A)
Special Sub-case: Events A and B are independent if P(B|A) = P(B)
For independent events:
P(A and B) = P(A) • P(B)
Examples:
Example 1: Two cards are drawn without replacement from a shuffled deck of 52 cards.
Event A: the first card is a face card
Event B: the second card is a face card
a) Determine the probability of P( A and B) .
b) Determine the probability of P( A and B) .
Example 2: A die is rolled three times. What is the probability of getting
a) a 4 on all three rolls?
b) at least one 4 on the three rolls?
Example 3: Two cards are drawn from a shuffled deck of 52 cards. What is the probability that
a) both cards are hearts?
b) neither card is a heart?
c) exactly one of the two cards is a heart?
d) both cards are aces?
Practice:
1. Determine if the events in each situation are dependent or independent.
a.
drawing a queen from a deck of cards then drawing another queen, with no replacement.
b.
drawing a king for the first card and a jack for the second card, with replacement.
c.
tossing a coin and getting a head, then rolling a die and getting 3.
d.
rolling a die and getting 6, then rolling it again and getting 1.
e.
tossing a coin 10 times and getting a head each time.
f.
winning at a solitaire game twice in a row.
2. Two cards are drawn without replacement from a shuffled deck of 52 cards. Determine the probability
of each event:
a. The first card is a heart and the second card is the queen of hearts.
b. The first card is the queen of hearts and the second card is a heart.
3. Two cards are drawn without replacement from a shuffled deck of 52 cards. Determine P(A and B) if
Event A: the first card is a spade
Event B: the second card is a face card
4. A bag contains 3 white balls and 2 black balls. The first draw produced a white ball. This first ball is
not replaced. What is the probability that the second draw will produce another white ball?
5. Three cards are drawn without replacement from a shuffled deck of 52 cards. What is the probability of
a. drawing 3 hearts
b. drawing 3 jacks
c. drawing 3 face cards
6. A biased coin with P(heads)
0.7 was tossed 3 times. The coin came up heads first, then tails twice.
What is the probability of this outcome?
7. Twenty percent of teachers catch the flu every year. Fifty percent of teachers have yearly flu shots. Ten
percent of teachers who have had flu shots get the flu. Are getting a flu shot and getting the flu
independent events?
8. In one bag, there are 2 white balls and 3 red balls. In a second bag, there are 4 white balls and 5 green
balls. One ball is drawn from each bag. What is the probability of
a. drawing 1 red ball and 1 green ball
b. drawing 1 red ball and 1 white ball
c. drawing 1 white ball and one green ball
d. drawing 2 white balls
9. The probability that you are late for the bus is 0.70. The probability that the bus is late is 0.10. What is
the probability that you are late and so is the bus?
10. What is the probability that a randomly selected person was born on the same day of the week as you
and is of the same gender?
11. Two cards are drawn from a shuffled deck of 52 cards without replacement. Determine the probability
of each event:
a. the cards are both spades
b. neither card is a spade
c. exactly one of the two cards is a spade
12. The photocopiers at a certain school are not very reliable. The probabilities that copier A, B, and C are
1 1
1
broken down are , , and respectively. What is the probability of each event?
3 2
6
a. all three copiers are broken down
b. no copier is broken down
c. only one copier is broken down
Assignment: EXERCISES 8.4: The Event „A and B‟
EXERCISES 8.4:
the event ‘A and B’
1) Classify the following events as dependent or independent.
a) Height and Weight of students:
i) Event 1 – the student is greater than 1.8 m tall
ii) Event 2 – the student weights more than 70 kg
b) Choosing two cards with replacement from a standard deck
i) Event 1 – first card is a Jack
ii) Event 2 – second card is a Queen
c) Choosing two cards without replacement from a standard deck
i) Event 1 – First card is a seven
ii) Event 2 – Second card is a seven
d) Rolling a die and rolling the die again
i) Event 1 – Number on the first row is a six
ii) Event 2 – Number on the second row is a two
1
1
3
2) Consider the two events such that P( A)
, P( B)
, and P( A or B)
,
2
3
4
a) Are A and B mutually exclusive?
b) Are A and B independent events?
1
2
1
3) Consider the two events such that P( A)
, P( B)
, and P( A and B)
.
4
5
10
a) Are A and B mutually exclusive?
b) Are A and B independent?
4) Let A and B be events with P( A) 0.6 , P( B) 0.4 , and P( A and B) 0.3 . Find:
a) P( A B)
b) P( B A)
c) P( A or B)
5) Let A and B be events with P ( A)
3
, P( B)
8
5
, and P( A or B)
8
3
. Find:
4
a) P( A B)
b) P( B A)
6) A red die and a blue die are tossed. What is the probability that the red die shows a 1 and the blue die
shows a 5 or 6?
7) Two cards are drawn with replacement from a standard deck of 52 cards. Determine the probability of :
a) Both cards are spades
b) Both cards are sevens
c) Neither card is red
d) The first card is a club and the second card is a diamond
e) One of the cards is red, and the other card is black.
8) Determine the probability of the question # 7, but this time without replacing the first card before
picking the second card.
9) The probability that Sara will pass grade 12 math and grade 12 physics this semester are 0.85 and 0.75
respectively. If these events are independent, what is the probability (to 4 decimal places) that she will
pass:
a) Both math and physics
b) Pass math but not physics
c) Pass physics but not math
d) Neither math nor physics
10) In a city school, 60% of students have blue eyes, 55% have dark hair, and 20% have neither blue nor
dark hair.
a) Find the probability that a random selected student will have blue eyes and dark hair.
b) State with a reason if these two characteristics are independent.
11) Consider two events such that P( A) 0.3 , P( B) 0.4 , and P( A or B) 0.58 . What can you say about
event A and B?
12) A die is rolled and a coin is tossed. What is the probability that the die shows an even number and the
coin shows up tails?
13) Four people roll a fair die to see who starts a game of scrabble. What is the probability that the first
person throws a 2, the second throws a 2, the third throws a 4, and the fourth person throws a number
other than a 2 or 4?
14) A triangular spinner has 3 equal sections, red, blue, and green. What is the probability (nearest
hundredth) of the spinner NOT landing on red on four consecutive spins?
Answer Key
1)
2)
3)
4)
a) dependent b) independent
c) dependent d) independent
a) No
b) No
a) No
b) Yes
a) 0.75
b) 0.5
c) 0.7
2
2
5) a)
b)
5
3
1
6)
18
1
1
1
1
1
7) a)
b)
c)
d)
e)
16
169
4
16
2
13
26
1
1
25
8) a)
b)
c)
d)
e)
204
51
17
221
102
9) a) 0.6375
b) 0.2125
c) 0.1125
d) 0.0375
10) a) 0.35
b) no, since P( B and D) P( B) P( D)
11) Not mutually exclusive, independent
1
12)
4
1 1 1 2
13)
6 6 6 3
14) 0.20
Math 12
8.5 Conditional Probability
P( B | A)
P( A and B)
P( A)
One bag contains 4 white balls and 6 black balls. Another bag contains 8 white balls and 2 black balls. A coin
is tossed to select a bag, then a ball is randomly selected from that bag.
a) What is the probability that a white ball will be drawn?
b) Suppose a white ball was drawn. What is the probability that it came from the first bag?
Example 1: A jar contains black and white marbles. Two marbles are chosen without replacement. The
probability of selecting a black marble and a white marble is 0.34, and the probability of selecting a black
marble on the first draw is 0.47. What is the probability of selecting a white marble on the second draw, given
that the first marble drawn was black?
Example 2: A company has two factories that make computer chips. Suppose 70% of the chips come from
Factory 1 and 30% come from Factory 2. In Factory 1, 25% of the chips are defective; in Factory 2, 10% of the
chips are defective.
a) Suppose it is not known from which factory a chip came. What is the probability that the chip is
defective?
b) Suppose a defective chip is discovered. What is the probability that the chip came from Factory 1?
Example 3: A new medical test for glaucoma is 95% accurate. Suppose 0.8% of the population has glaucoma.
What is the probability of each event?
a) A randomly selected person will test negative.
b) A person who tests negative has glaucoma.
c) A person who tests positive does not have glaucoma.
Assignment: EXERCISES 8.5: Problems Involving Conditional Probabilities
EXERCISES 8.5
Problems Involving Conditional Probabilities
1) Two cards are drawn with replacement from a deck of cards. Calculate the probability that two queens
are drawn given that at least one queen is drawn.
2) An octagonal die numbered 1 – 8 is rolled twice. What is the probability of rolling only one 8, given
that at least one 8 is rolled?
3) The table shows the distribution of blood types for students in the first year at a local college.
O
A
B
AB
Total
Male (M)
210
174
74
42
Female (F)
315
261
111
63
Total
a) Complete the total in the chart
b) How many students are in the first year at the college?
c) If a student is selected at random, determine the probability (to four decimals) that the student:
i) Is male
ii) Has blood type A
iii) Is male and has blood type A
d) Are the events “the student is male” and “the student has blood type A” independent of each other?
Why?
e) Determine the probability (to four decimals) that:
i) A female student has blood type B
ii) A female student does not have blood type O
iii) A student with blood type A is male
iv) A student with blood type AB is female
4) Packet 1 contains 8 red balloons and 4 blue balloons. Packet 2 contains 5 red balloons, 3 green balloons
and 4 blue balloons. A die is rolled. If the result is a one or a six then a balloon is chosen at random
from packets 1. Otherwise, a balloon is chosen from packet 2.
a) Blue
b) Green
5) In a diagnostic test for a disease, sometimes a positive reaction is obtained, even when the disease is not
present. In the past the test has given a positive reaction to 90% of people having the disease and to 5%
of people who do not have the disease. If 10% of the population has the disease, calculate the
probability that a person chosen at random would not react positively to the test.
6) Three machine A, B, C produces respectively 50%, 30%, 20% of the items produced daily by a
manufacturing company. The percentages of defective items produced by the machines are respectively
5%, 2%, and 1%. What is the probability that an item selected at random from the daily output is
defective?
7) A golfer is practicing putting. The probability that he holes the first putt is 0.4. Each time he holes a
putt the probability that he holes the next putt increases by 25%. Each time he misses a putt the
probability that he misses the next putt increases by 20%. Calculate the probability that:
a) He holes the first two putts
b) He misses the first two putts
c) He holes two of the first three putts
8) A box contains 10 cans of cola and 6 cans of lemonade. A second box contains 8 cans of cola and 8 cans
of lemonade. One of the boxes is chosen at random and a can is selected at random from that box. The
selected can is a can of lemonade. A second can is then selected from the same box. Determine the
probability that the second can is also lemonade.
9) A packet of candy contains 10 individually wrapped pieces of candy, each of which is either orange or
lemon flavored. One particular packet contains 8 orange and 2 lemon candies and a second packet
contains 7 orange and 3 lemon candies. One of these packets is chosen at random and a candy is
selected at random from that packet. It is lemon flavored. A second piece of candy is then selected
from the same packet. Calculate the probability that the second candy is orange flavored.
Answer Key:
1
25
14
2)
15
3) b) 1250
c) i) 0.4000 ii) 0.3480
iii) 0.1392
d) Independent since P( M and A) P( M ) P ( A)
1)
4)
5)
6)
7)
8)
9)
e) i) 0.1480 ii) 0.5800
1
1
a)
b)
3
6
0.865
0.033
a) 0.2
b) 0.432
43
105
37
45
iii) 0.4000
c) 0.2138
iv) 0.6000
Math 12
8.6 Probability and Combinatorics
For experiments involving many possible outcomes we turn to combinatorics to determine the number of
favourable outcomes and possible number of outcomes. Yes, our first unit is coming back to haunt you!
Example 1: Suppose two cards are drawn without replacement from a shuffled deck of 52 cards. Calculate the
probability that they are both aces.
Determine the number of favourable outcomes (selecting 2 aces):
Determine the number of possible outcomes (selecting 2 cards):
P(both cards aces) =
Example 2: Three prizes are awarded for best costume at Van Tech‟s Halloween Dance. There are one hundred
students who dressed up. (Assume all costumes have the same chance of winning a prize.)
a) What is the probability that Larry wins 1st prize, Moe wins 2nd, and Curly wins 3rd?
b) What is the probability that Larry, Moe, and Curly are the 3 prize winners?
Example 3: In the BC49 draw 6 numbered balls are randomly selected without replacement from 49 balls
(numbered 1 to 49). You have a ticket with your 6 numbers.
a) What is the probability that all 6 of your numbers are drawn? (Hey, you never know!)
b) What is the probability that only three of your numbers are drawn?
c) Are you more likely to win the lottery, or get hit by lightning?
Example 4: The students in a grade 12 math class somehow manage to convince the substitute that their teacher
lets them play cards during class time. Michelle draws five cards without replacement from a shuffled deck of
52 cards. Determine the probability of each of the following events.
a) The five cards are all spades.
# of hands of 5 spades:
# of hands of 5 cards:
P(5 spades) =
b) There are 3 spades and 2 hearts.
c) There are 3 spades and 2 non-spades.
d) There are only 2 spades in the 5-card hand.
e) There are 2 aces and 2 kings and one other card of a different type.
Assignment: EXERCISES 8.6: Problems Involving Permutations and Combinations
EXERCISES 8.6
Problems Involving Permutations and Combinations
1) A marble is drawn at random from a box containing 10 red, 30 yellow, 20 blue, and 10 pink marbles.
Find, as an exact fraction, the probability that the marble drawn is:
a) Yellow or red
b) Not blue
c) Green
d) Red, pink or blue
2) A bag contains 6 blue marbles and 10 yellow marbles. Two marbles are drawn from the bag without
replacement.
a) Use the multiplication law to determine:
i) P(both are blue)
ii) P(first is blue and second is yellow)
iii) P(one is blue and one is yellow)
b) Use combinations or permutations to determine:
i) P(both are blue)
ii) P(first blue and second is yellow)
iii) P(one is blue and one is yellow)
3) Two marbles are drawn at random from a box containing 10 red, 20 yellow, 15 blue and 15 pink
marbles. Find the probability (nearest thousandth) that:
a) Both marbles are red
b) The first marble is red and second is yellow
c) Neither marble is pink
d) Both marbles are either blue or red
4) In a card game you are dealt 5 cards from a pack of 52 well-shuffled cards. What is the probability,
expressed in combination notation, that you have:
a) Four queens?
b) One face card?
c) A, 2, 3, 4, and 5?
d) At most one King?
e) At least one diamond?
5) Three prizes are rewarded in a raffle during a halftime show at a school basketball game. Ben Janelle,
Jamie, and 17 other students each have one ticket.
a) If the raffle has a first, second, and third prize, what is the probability (in exact value), that Ben wins
first prize, Janelle wins second prize, and Jamie wins the third prize?
b) If the raffle has three identical prizes, what is the probability that Ben, Janelle, and Jamie win the
prizes?
6) Three toys are selected at random from a box of nine toys, three of which are defective. Find the
probability (exact value) that the toys selected:
a) None are defective
b) There are more defective than non-defectives
7) A bank card personal identification number consists of any four digits. Repeat digits are allowed and
the code can start with a zero. What is the probability that a code begins and ends with 5?
8) In a small rural town there is only one football field to be shared by the five local teams. In the league
completion, each team plays two games against every other team. What is the probability that the fist
game scheduled will have last year‟s two best teams in opposition?
9) A jar contains 32 balls, equal number of red, blue, green and yellow. Five balls are drawn from the jar,
simultaneously and at random. Determine the probability (nearest thousandth) of each event:
a) There are three red balls and two yellow ones
b) The balls are three of one colour and two of another colour
c) All five balls have the same colour.
10) There are twelve boys and ten girls in an English class. A yearbook committee of four is chosen at
random.
a) What is the probability (nearest hundredth) that Ryan, the school present, and three girls are on the
committee?
b) If Ryan must be on the committee, what is the probability (nearest hundredth) that three girls are
chosen?
11) A six digit numeral is represented by the digits 1, 2, 3, 4, 5, and 6 in any order. If one of the three
numerals is chosen at random what is the probability that:
a) All the digits are in decreasing order?
b) The even digits are in decreasing order?
12) Chandra buys two of a total of thirty raffle tickets. There are two winning numbers. What is the
probability that she wins exactly one prize?
13) The letters of the word OKOTOKS are arranged. What is the probability (nearest hundredth) that an
arrangement, chosen at random from all possible arrangement, begins and ends with a vowel?
14) The girls soccer team consists of 11 starters and 5 substitutes. The probability (nearest hundredth) that
at least 2 of the 16 girls have the same birthday is _____________
Answer Key:
1) a)
4
7
4) a)
( 4 C4 )( 48 C1 )
52 C5
5) a)
1
6840
9) a) 0.008
11) a)
1
720
b)
5
7
c) 0
d)
b)
b)
1
1140
1
6
1
1
ii)
8
4
2) i)
( 12 C1 )( 40 C4 )
52 C5
b) 0.093
b)
4
7
6) a)
c)
5
21
c) 0.001
12)
56
435
iii)
1
3) a) 0.025
2
( 12 C1 )5
52 C5
b)
d)
19
84
b) 0.056
( 4 C0 )( 48 C5 )
52 C5
7)
1
100
10) a) 0.02
b) 0.09
13) 0.14
14) 0.28
c) 0.559 d) 0.169
( 4 C1 )( 48 C4 )
52 C5
8)
1
10
e) 1
C5
52 C5
39
Math 12
The BIRTHDAY PROBLEM
What is the probability that there are at least two people in this class who share the same birthday?
Practice:
1. What is the probability that on a soccer team with 24 players,
a) at least two will have the same birthday?
b) when asked to choose a number from 1 to 50, inclusive, at least two will choose the same number?
2. At a Christmas dinner, each of the 25 guests is asked to choose one of 5 appetizers, one of 4 soups, one
of 3 main courses, and one of 5 deserts. What is the probability that at least two of the guests will choose the
same meal?
3. On the show „Deal or No Deal‟ a contestant must open one of 12 remaining suitcases. You and two
friends are watching and each of chooses a case without telling the others. What is the probability that at
least two of you (including the contestant) choose the same suitcase?
Answers: 1. a) 0.5383
b) 0.9987
2. 0.6424
3. 0.4271
Math 12 8.7 Calculating Probabilities with the Binomial Theorem
Picture a die being rolled. Use a probability tree to determine the probability of correctly guessing the
outcome in exactly 2 out of 3 rolls of a die.
Probability tree:
1st roll
2nd roll
3rd roll
P(exactly 2 correct guesses) =
What is the probability of guessing the outcome in exactly 2 out of 6 rolls of a die?
What is the probability of guessing the outcome in exactly x out of 6 rolls of a die?
Example 1: A fair coin is flipped 4 times. Calculate the following:
a) P(exactly 2 tails) =
b) P(at least 3 tails) =
Example 2: A fair coin is flipped 12 times. Calculate the following:
a) P(exactly 6 tails) =
b) P(exactly 2 tails) =
c) P(exactly x tails) =
Example 3: A math 12 student arrives on the day of the test without having studied and must answer each
question by randomly guessing. There are 10 multiple choice questions and each question has 4 choices (of
which only one is correct). Determine the probability of…
a) guessing correctly exactly twice.
b) guessing correctly at least twice.
c) guessing correctly exactly x times.
Assignment: Exercises 8.7: Calculating Probabilities with the Binomial Theorem
Exercises 8.7
Calculating Probabilities with the Binomial Theorem
1. In history class, Colin and Diana both write a multiple-choice quiz. There are ten questions. Each
question has five possible answers. What is the probability that
a. Colin will pass the test (score at least 5) if he guesses an answer to each question?
b. Diana will pass the test if she studies so that she has a 75% chance of answering each question
correctly?
2. The manufacturing sector contributes 17% of Canada‟s gross domestic product. A customer orders 50
components from a factory that has a 99% quality production rate (99% of the products are defect-free).
Find the probability that
a. none of the components in the order are defective
b. there is at least one defective product in the order
c. there are at least two defective products in the order
3. Approximately 3% of the eggs in a supermarket are cracked. If you buy two dozen eggs, what is the
probability that
a. none of your eggs are cracked?
b. at least one egg is cracked?
c. exactly two eggs are cracked?
d. What must you assume?
4. A pair of dice is rolled 20 times. What is the probability that double sixes are rolled
a. exactly twice?
b. more than once, but no more than three times?
5. At the 1999 Pan American Games, Kim Eagles won gold for Canada in the women‟s air pistol event.
Max averages a 60% hit rate when shooting pop cans with this air rifle. He fires ten shots at a row of
pop cans. Estimate the probability that he will hit
a. exactly eight cans
b. at least five cans
c. at least eight cans
d. Draw a Venn Diagram showing the events of parts (b) and (c)
e. Estimate the probability that he will hit at least five, but no more than seven, cans.
6. On Alan‟s journey to work, he has to pass through 1 intersections with the traffic lights that operate
independently. The chance that any set is green when he reaches it is 0.5. If he is stopped at fewer than
three sets of lights, he has time for a coffee before work. But if he is stopped at more than 8 sets of
lights, he is late. If Alan is late more than twice in a five-day work week, he has to see his manager.
What is the probability that
a. Alan has time for the coffee on any given day?
b. Alan has to see the manager in any given week?
5 0
.8
5 1
2
.7
5
7. Lina was told that Vince Carter makes 85% of his free-throw attempts. Since 1
, Lina
expects that in 15 attempts Carter will make 13 baskets. Is Lina‟s reasoning correct? Which is more
likely, that Carter makes exactly 13 baskets or Carter does not make exactly 13 baskets?
Answer Key:
1. a) 0.033 b) 0.980
3. a) 0.481 b) 0.519 c) 0.127 d) independence
5. a) 0.121 b) 0.834 c) 0.167 e) 0.666
7. Compare the probability that he will make the
likely?
2. a) 0.605 b) 0.395 c) 0.089
4. a) 0.088 b) 0.103
6. a) 0.019 b) 0.003
basket to the probability that he will miss. Which is more
Review
1. What is the probability that a card drawn at random from a standard deck is
a. an ace?
b. a face card from either hearts or diamonds?
2. What is the probability that a factor of 24, selected at random, is
a. even?
b. a prime number?
In questions 3 to 6, list a sample space for the experiments and determine the probability of the required event.
3. When tossing a fair coin and cutting a standard deck of cards, what is the probability of obtaining a head
and a spade card?
4. When rolling a fair six-sided die and cutting a standard deck of cards, what is the probability of
obtaining a two and an ace?
5. In the random selection of a movie (comedy, drama, adventure, or science fiction) and a snack (chips,
chocolate or popcorn), what is the probability of choosing a comedy with chips?
6. Randomly selecting a time (17:00, 18:00 or 19:00) and a physical activity (jogging, rollerblading,
cycling or swimming), what is the probability of choosing swimming at 17:00?
7. Two fair six-sided dice are rolled.
a. List a sample space for this experiment in table form.
b. Determine the probability that the sum of the two dice is a prime number.
8. A fair coin is tossed five times in a row. What is the probability of getting at least one head?
9. Two fair, eight-sided dice are rolled. What is the probability of obtaining a sum of 13 or less?
10. Two numbers are relatively prime if they have no common prime factors.
a. List a sample space in the experiment of rolling two dice.
b. Identify the outcomes that are relatively prime.
c. Determine the probability of rolling two numbers that are relatively prime.
Classify the events in questions 11 to 15 as either independent or dependent.
11. The experiment is to check the training of employees and the productivity of a company. The first
event is that the employee is well trained and the second event is that the company meets its production
quota.
12. The experiment is to observe the sky and the season of the year. The first event is that the sky is cloudy
and the second event is that the season is winter.
13. The experiment is to cut a deck of cards and to roll two dice. The first event is that a club is chosen and
the second event is that a pair is rolled.
14. The experiment is to check the handedness of a student and to check the hair colour. The first event is
that the student is left-handed and the second event is that the student has blonde hair.
15. The experiment is to study the athletic activities of a person. The first event is that the person plays
squash and the second event is that the person plays tennis.
Classify the events in questions 16 to 19 as either mutually exclusive or not mutually exclusive.
16. The experiment is rolling a die. The first event is that the number is greater than 3 and the second event
is that the number is even.
17. The experiment is choosing a single activity for Saturday evening. The first event is to watch a video at
home and the second event is to go dancing.
18. The experiment is answering a multiple-choice question. The first event is that the correct answer is
chosen and the second event is that answer A is chosen.
19. The experiment is cutting a deck of cards. The first event is that the card is a face card and the second
event is that the card is less than 7.
20. Sixty percent of young drivers take professional driver training. In their first year of driving, 25% of
young drivers have an accident.
a. Assume that the events, taking driver training and having an accident, are independent. Draw the
probability tree.
b. Based on part (a), what is the probability that someone who has taken driver training will have an
accident in the first year of driving?
c. Studies show that 10% of those who do take driver training have an accident in their first year.
Is your result from part (b) supported by the studies?
d. What does this tell you about the assumption made in part (a)? What advice would you give to
someone who has just got a learner‟s permit?
21. The distribution of blood types in North America is
type O: 44%
type A: 42%
type B: 10%
type AB: 4%
a.
b.
c.
d.
Are the blood types mutually exclusive? Defend your decision using the data given.
What is the probability that a couple both have type O blood?
What is the probability that neither person in a couple has a type AB blood?
When would you not consider blood types of two people independent?
22. There are three traffic lights along the route that Kia takes from home to school. The first traffic light is
red 50% of the time. The second light is red 40% of the time, and the third light is red 60% of the time.
The lights operate on separate timers. What is the probability that all three lights will be red as Kia reaches
them on her way to school?
23. Three of the top Canadian paid-circulation magazines are Reader‟s Digest (Canadian Edition),
Chatelaine, and Maclean‟s. A market survey has estimated that the probability of a household‟s subscribing
to these magazines is:
Subscription
a.
b.
c.
Probability
Reader‟s Digest
0.6
Chatelaine
0.5
Maclean‟s
0.4
Reader‟s Digest and Chatelaine
0.2
Reader‟s Digest and Maclean‟s
0.25
Chatelaine and Maclean‟s
0.15
All three
0.05
What is the probability that a household chosen at random subscribes to only Reader‟s Digest?
What is the probability that a household chosen at random subscribes to Chatelaine, Maclean‟s, or
both?
What is the probability that a household chosen at random subscribes to one magazine only?
A committee of five people is randomly selected from ten females and eight males.
24. What is the probability that there are exactly three females on the committee?
25. What is the probability that there are exactly three males on the committee?
26. What is the probability that there are exactly two females on the committee?
Three people line up to buy a ticket at the movie theatre.
27. What is the probability that they line up in descending order of age?
28. What is the probability that they line up in ascending order of age?
At a school party, ten students sit randomly around a circular table that seats ten people.
29. What is the probability that Janna is sitting next to Roberto?
30. What is the probability that Janna and Roberto are sitting opposite each other?
31. What is the probability that Janna and Roberto are not sitting opposite or beside each other?
32. A card trick involves the drawing of aces from a well-shuffled deck of 54 cards, including jokers.
a. What is the probability of cutting an ace from a full deck of cards?
b. What is the probability of drawing out four aces in four draws from the deck, without
replacement?
33. A lock on a briefcase has three wheels, each labeled from 0 to 9. What is the probability of a person‟s
guessing the correct entry code, if
a. there are no repeated digits in the code?
b. the code may have repeated digits?
34. Two sisters, Zana and Kari, are among 15 girls trying out for forward positions on a hockey team. On
the first day of practice, the coach randomly sets up five forward lines of three players each. If the coach
differentiates between the positions of right wing, left wing, and centre,
a.
b.
c.
d.
how many ways can the coach choose the first line? The second line?
how many ways can the coach set up the five lines?
what is the probability that Zana and Kari are on the same line?
what are the answers to parts (a), (b), and (c) if the coach does not differentiate between the three
positions on the line?
35. In the game of bridge, four players are dealt 13 cards each from a well-shuffled deck of 52 playing
cards.
a. What is the probability that one of the players is dealt all the spades?
b. What is the probability that each of the four players receives a complete suit in his or her hand?
c. What is the probability that any player holds all the aces?
d. What is the probability that the player who deals does not have a face card, or an ace, in his or
her hand?
36. Matching the world typing record of 176 words/min would be difficult even if one started early in life.
If an infant types three keyboard strokes and all the characters are letters of the alphabet, what is the
probability that
a. the first letter typed is a vowel?
b. all three characters typed are vowels?
c. the three letters typed are different, and in alphabetical order?
37. The dice game of Yahtzee® uses five regular dice. A yahtzee occurs when you roll the same number
on each of the dice. What is the probability that a single roll of all of the dice yields
a. a yahtzee of sixes?
b. any yahtzee?
c. not even one pair of dice with the same number?
38. Sir John A. MacDonald, Canada‟s first prime minister, and Jean Chretien, Canada‟s twentieth prime
minister, were both born on January 11.
a. What is the probability that at least two of the next ten Canadian prime ministers will have been
born in the same month? What simplifying assumption are you making?
b. What is the probability that at least two of the next ten Canadian prime ministers will have the
same birthday?
c. What is the probability that at least two of the next twenty Canadian prime ministers will have
the same birthday?
39. A small town has a network of 115 residential streets, each containing approximately the same number
of residents. A canvasser randomly selects twenty names from the phone book.
a. What is the probability that at least two of the people selected live on the same street?
b. Why is the information “approximately the same number of residents” relevant?
c. How many people must the canvasser sample to have at least a 75% chance of having a
minimum of two on the same street?
40. The percent of Canadian families with five children decreased from 3.9% in 1961 to 0.4% in 1991.
Suppose a family is planning to have five children. If a male child and female child are equally likely, what
is the probability that the children are
a. two boys, one girl, two boys, in that order?
b. all boys?
c. one girl, three boys, one girl, in that order?
d. Is it a reasonable assumption that a newborn is equally likely to be a male or female? Is the
gender of a child independent or that of a previous child? Research and discuss.
41. A bridge hand of 13 cards is called a „Yarborough‟ when it contains to tens, no aces and no face cards.
a.
b.
c.
d.
Is the dealing of a bridge hand a sequence of independent or dependent events?
What is the probability of being dealt a Yarborough?
What is the probability of not being dealt a Yarborough?
How many hands would you expect to be dealt before being dealt a Yarborough?
Determine the conditional probability in questions 42 to 45. Round answers to three decimal places where
necessary.
(A
a
n
dB
) 0
.5
42. Given: P
PA
( )
0.75
Find: P(B | A) .
(
e
v
e
n
a
n
d
d
i
v
i
s
i
b
l
e
b
y
5
) 0
.
1
43. Given: P
P
(even) 0
.4
(
e
v
e
n
a
n
d
d
i
v
i
s
i
b
l
e
b
ye
5
|v
e
n
)
Find: P
.
(
b
l
o
n
d
e
a
n
d
b
l
u
e
e
y
e
s
) 0
.
2
44. Given: P
P
(
b
lu
e
e
y
e
s
) 0
.2
5
(
b
l
o
n
d
e
a
n
d
b
l
u
e
e
y
e
s
|
b
l
u
e
e
y
e
s
)
Find: P
.
(
r
e
d
a
n
d
s
i
z
e
m
e
d
i
u
m
) 0
.
3
45. Given: P
P
(
s
i
z
e
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Find: P
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46. Find the probability of having two boys, given that the first child is a boy. Assume that the gender of a
child is independent of its sibling‟s gender and that male and female births are equally likely.
47. Find the probability of drawing two cards from a deck of 52 cards and obtaining heart cards both times,
given that a heart is drawn the first time.
48. The results of a recent census on the number of children in families in Canada are:
no children
40%
1 child
25%
2 children
18%
3 or more
17%
If a Canadian family is selected at random, what is the estimated
probability that
a. there are more than two children?
b. there are more than two children, given that the family has at
least one
child?
49. At a certain university, 75% of the first-year engineering students passed mathematics, 85% passed
chemistry, and 90% passed mathematics or chemistry. A first-year engineering student is selected at
random.
a. Determine the probability that the student passed mathematics, given that the student passed
chemistry.
b. Determine the probability that the student passed chemistry, given that the student passed
mathematics.
50. Polygraph or lie detector tests are generally not very accurate. Suppose a polygraph is 70% accurate,
that is, if you lie, the polygraph indicates so 70% of the time, and if you tell the truth, the poly graph
indicates so 70% of the time.
a. Assume that the average person is truthful 90% of the time. Determine the probability that a
person is telling the truth, given that the polygraph test indicates truth.
b. Assume that criminals are truthful 15% of the time. What is the probability that a criminal is
lying, given that the polygraph test indicates truth?
51. Suppose that a breathalyzer test is 95% accurate. Also assume that the drivers the police ask to take the
test have drunk too much alcohol 80% of the time.
a. Determine the estimated probability that a driver is falsely charged with impaired driving, that is,
the driver is not impaired given that the breathalyzer tests over the limit.
b. After a driver tests positive on a breathalyzer, the police generally order a blood test. Why?