Download Section 7-1 – How Probabilities are Determined

Section 9-1 – How Probabilities are Determined
Experiment: An activity whose results can be observed and recorded.
Each possible result of an experiment is called an outcome.
Sample Space: The set of all possible outcomes for an experiment.
Example:
What is the sample space for flipping a coin one time?
What is the sample space for rolling a die?
Event: Any subset of a sample space is an event
Ex: Rolling an odd # {1, 3, 5}
Example: Suppose we toss a fair coin 3 times and record the results.
a) Find the sample space for this experiment
b) Find the event A of tossing no heads
c) Find the event B of tossing a tail on the first toss
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Determining Probabilities
Experimental or Empirical Probability: When a probability is determined by observing
outcomes of experiments. You actually perform the trials or experiments and record the results as a
probability.
P(A) denotes the “probability of event A occurring”
Law of Large Numbers (Bernoulli’s Theorem): If an experiment is repeated a large number of
times, the experimental probability of a particular outcome approaches a fixed number as the
number of repetitions increases.
The fixed number mentioned above is “theoretical probability”
We assign theoretical probabilities to the outcomes under ideal conditions. For example, in a
perfect world we would expect to roll a 4 on a die 1 out of every 6 times so P(rolling a 4) = 1/6.
Note that if we actually rolled a die several times we might NOT roll the number 4 one out of every
6 times. Maybe we rolled the 4 three out of 7 times. In that case the experimental probability
would be 3/7 even though the theoretical was 1/6. The Law of Large Numbers says that if we roll
the die more and more times, the experimental probability will get closer and closer to the
theoretical probability.
When one outcome is just as likely as another (as in coin tossing or rolling a die), the outcomes are
called equally likely.
Probability Rules:
An impossible event has probability ________.
An event certain to occur has probability ________.
For any event A, ____ ≤ P(A) ≤ ____ .
Example: Let set A = {1, 2, 3, 4, 5, 6, 7, 8}.
Calculate the probability for each of the following events:
a) A = an even number is drawn
b) B = a number more than 5 is drawn
c) C = a prime number is drawn
d) D = an even number that is a multiple of three is drawn
e) E = drawing either a prime or a composite number
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Example: In a movie theater, 50 people have popcorn and 35 people do not. What is the
probability that a randomly selected person has popcorn?
Example: A multiple choice test has 5 choices for each problem. What is the probability that if you
guess, you will guess incorrectly?
Example: Roll 2 dice and record their sums in the following table.
a) What is the probability of getting a sum of 9?
b) What is the probability of getting a sum of 7?
+
1
2
3
4
5
6
1
2
3
4
5
6
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Venn Diagrams:
INTERSECTION: P(A and B) = P( A  B)
UNION: P(A or B) = P( A  B)
P(A and B) = P(A and B both occur)
P(A or B) = P(A occurs or B occurs or both)
When finding the probability that event A occurs OR event B occurs, find the total number of ways
that A can occur and the number of ways B can occur, but find the total in such a way that no
outcome is counted more than once!!
The Addition Rule:
P( A  B)  P( A)  P( B)  P( A  B)
Example: There are 30 students in a shop class. 18 build bird houses, 8 build mailboxes, 3 build
both. Set up a Venn Diagram to illustrate.
Example: If one student is randomly selected, calculate P(birdhouse OR mailbox).
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Example: A single card is drawn from a deck of cards.
a) Find the probability of selecting a Jack OR a black card.
b) Find the probability of selecting a Five OR a heart.
c) Find the probability of selecting a Face card OR a red card.
d) Find the probability of selecting a Queen AND a club.
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Mutually Exclusive – Events A and B are mutually exclusive if they have no elements in common.
ie, A  B  
If two events are mutually exclusive, then P( A  B) = ______.
So for 2 disjoint events, P( A  B) =
Example: A roulette wheel has 38 slots (0, 00, 1–36).
Suppose you bet on the number 18.
a) P(win) =
b) P(lose) =
c) P(lands on 9 or 26) =
The Complement of A – All outcomes in which A does not occur. Denoted A .
Properties:
P(A) + P( A ) = _____
P( A ) = 1 – _____
Example: In reality, when a baby is born, P(boy) = 0.512. Find the probability of having a girl.
Example: In a survey of 500 randomly selected high school students, it was determined that 420
played organized sports. What is the probability that a randomly selected student does not play
organized sports.
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Section 9-2 – Multistage Experiments (Tree Diagrams & Geometric Probabilities)
Property: The Multiplication Rule for Probabilities on Tree Diagrams
For all multistage experiments, the probability of the outcome along any path of a tree diagram is
equal to the product of all of the probabilities along the path.
Example: Use a tree diagram to create the sample space for the gender of the children in a family
with three children.
Based on the tree diagram, calculate the following probabilities
a) P(exactly one boy)
b) P(exactly two girls)
c) P(all boys)
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Independent Events: When the outcome of one event has no influence on the outcome of a
second event.
If two events, A and B, are independent, then
P( A  B) = P(A) · P(B)
Dependent Events: When the outcome of one event has DOES influence on the outcome of a
second event.
If two events, A and B, are dependent, then
P( A  B) = P(A) · P(B after A)
Classify the following events as independent or dependent:
a) Joe and Bob decide to ride the bumper cars at the fair. Joe picks a red car and then Bob picks a
green one.
b) Sue draws a jack out of a deck of cards, replaces it, and then draws a 7.
c) Tommy spins a spinner. It lands on green for his first spin and blue on his second spin.
d) Amy is a first grade teacher. She wants to randomly assign 2 prizes to her students so she draws
a name out of a hat, sets that name aside and then draws another name.
Example: Create the sample space for the following scenario:
Answer test question #1 (true/false), then answer test question #2 (multiple choice w/ 5 choices)
What is the probability of answering “true” on #1 and “B” on #2 ?
P(true and B) =
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Example: An assembly line has two inspectors. The probability that the first inspector will miss a
defective item is 0.05. If the defective item passes the first inspector, the probability that the second
inspector will miss it is 0.01. What is the probability that a defective item will pass by both
inspectors?
Example: You choose 2 cards from a standard deck of cards without replacement.
Calculate P(both cards are Kings) =
Example: A bag contains 4 blue marbles and 2 red marbles.
a) P(1st is blue and 2nd is red and 3rd is blue)
(marbles not replaced)
b) P(1st is blue and 2nd is red and 3rd is blue)
(marbles replaced)
Example: The following are two boxes containing black and white balls. A ball is drawn at random
from box 1. Then, a ball is drawn at random from box 2, and the colors of the balls are recorded in
order. Find each of the following:
a) P(2 white balls) =
b) P(at least one black ball) =
c) P(one white ball and one black ball) =
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Example:
Consider the three boxes below. A letter is drawn from box 1 and placed in box 2. Then, a letter is
drawn from box 2 and placed in box 3. Finally, a letter is drawn from box 3. What is the
probability that the letter drawn from box 3 is B?
AAB
AB
ABBB
Example: In a certain population of caribou, the probability of being sickly is 1/20. If a caribou is
sickly, the probability of being eaten by a wolf is 1/3. If a caribou is not sickly, the probability of
being eaten by a wolf is 1/150. If a caribou is chosen at random, what is the probability that it will be
eaten by a wolf?
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Modeling Games
Example: Suzy and John are playing a card game. A box consists of 2 red cards and 2 black cards.
John randomly draws two cards without replacement. If the two cards are the same color, he wins.
Otherwise, Suzy wins. Does each player have an equal chance of winning?
Geometric Probability (Area Models)
Example: Consider the following diagram of an overhead view of several regions of land owned by
various farmers.
Farmer Clark
Farmer Lex
Farmer Olsen
Farmer Kara
Farmer Lane
If an object is dropped from a hot air balloon, what is the probability it lands on Farmer Lex’s land?
What is the probability it lands in either Farmer Clark, Farmer Olsen, or Farmer Lane’s land?
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Example: In a school, 50 students are taking math, 60 students are taking biology, and 30 students
are taking both. Suppose the school has 150 students. If a student chosen at random is known to be
taking math, what is the probability that he is also taking biology?
Section 9-3 Using Simulations in Probability
The Lotto-Choco candy company wants to promote their new candy bar, the Delish bar. They print
a D, E, L, I, S, or H inside the wrapper of each candy bar. Anyone who finds all 6 wrappers wins
100 free candy bars. Let’s assume that the same number of candy bars are printed with each letter
and that the candy bars are randomly distributed throughout the country (thus making it equally
likely that you would get any letter). Suppose Walter really likes the Delish bar, but is not sure
whether he wants to spend the money to really try to collect all 6 letters and win the 100 candy bars.
Could he design a simulation to predict or estimate the number of candy bars he would actually
have to purchase?
A simulation is a model or an experiment where we determine approximate results when it is too
time consuming or expensive to actually conduct trials. For a good simulation, the probabilities in
the experiment must be the exact same as the probabilities in the actual event you are simulating.
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Section 9-4 – Odds, Conditional Probability, and Expected Value
The idea of odds is similar to the idea of probability but with a different perspective. Remember
that probability was represented by a ratio of the ways a specific outcome could happen as
compared to the total possible outcomes. Odds is represented by a ratio of the ways a specific
outcome could happen as compared to the ways the outcome could NOT happen.
Definition of Odds:
Let P(A) be the probability that event A occurs and P ( A) be the probability that event A does NOT
P( A )
occur. Then the odds in favor of event A occurring are
and the odds against event A
P ( A)
occurring are
P( A)
.
P( A)
Let’s think of it using a probability we are familiar with.
What is the probability of rolling a 5 on a die?
What are the odds of rolling a 5 on a die? This is also known as the odds in favor of rolling a 5.
What is the probability of NOT rolling a 5 on a die?
What are the odds of NOT rolling a 5 on a die? This is also known as the odds against rolling a 5.
Although odds can be written as a fraction or with a :, I generally prefer to use the colon as a
reminder to myself that I’m working with odds and save the fraction for probability.
Example:
Think of a deck of cards and find the following probabilities and odds.
a) Probability of drawing a heart.
b) Odds of drawing a heart.
c) Probability of drawing a 7.
d) Odds of drawing a 7.
e) Odds against drawing a 7.
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Example:
A soft drink company has a promotion going where they put messages under the caps of a few
drinks to award prizes to the person who purchased that drink. The probability of winning specific
prizes are as follows:
Winning another drink: 1/20
Winning a bicycle: 1/1,000
Winning $1,000,000: 1/10,000,000
a) What are the odds of winning $1,000,000?
b) What are the odds of winning another drink?
c) What are the odds against winning another drink?
d) What are the odds against winning a bicycle?
e) What is the probability of NOT winning another drink?
f) What is the probability of NOT winning a bicycle?
Conditional Probabilities
The probability that event B will happen GIVEN that event A has already happened is:
P(B│A) =
P( A  B)
P( A)
Example: A spinner contains numbers from 1 – 10. You spin the spinner one time. Calculate the
following probabilities:
a) P(landing on 5 given the number is odd)
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b) P(landing on 5 given the number is larger than 3)
c) P(landing on 5 given the number is even)
Example: A bag contains 3 white marbles, 4 red marbles, and 2 green marbles. A marble is drawn
at random and NOT replaced. A second marble is drawn. Calculate the following probabilities:
a) P(second ball is green given that the first ball was white)
b) P(second ball is green given that the first ball was red)
c) P(second ball is green given the first ball was green)
Expected Value
Expected value is used to predict the average result of an experiment when it is repeated many
times. It cannot be used to determine the outcome of any single experiment.
In an experiment, if the possible outcomes are a1, a2, a3, …, an, with respective probabilities p1, p2,
p3, …, pn, then the expected value is:
E =  a1  p1    a2  p2    a3  p3   ...   an  pn 
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Example: You play a game where you bet $1. If you win the game, you win $20. If you lose the
game, you lose your dollar. The probability of winning the game is 0.08. Find your expected
winnings.
Is this a fair game? (ie, price of playing equals the expected winnings)
Example: Suppose 5 quarters, 5 dimes, 5 nickels, and 10 pennies are in a box. One coin is selected
at random. What is the expected value of this experiment?
Section 9-5 – Permutations and Combinations
Fundamental Counting Rule – If one event can occur in M ways, and another event can occur in N
ways, then the 2 events can occur together in M·N ways.
Example: How many license plates can be formed containing 3 letters followed by 3 numbers?
(repeats allowed)
_______ _______ _______ _______ _______ _______
(no repeats allowed)
_______ _______ _______ _______ _______ _______
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Factorials:
n! = n(n –1)(n – 2)(n – 3) · · · 3 · 2 · 1
Example:
5! =
10! =
Factorial Rule – N items can be arranged in N! different ways.
Example: How many different ways can 7 different video game cartridges be arranged on a shelf?
PERMUTATIONS – The number of ways that r items can be selected from n items in a specific
n!
order is:
nPr =
(n  r )!
(an arrangement of things in a definite order with NO repetitions)
Example: How many ways can a President, Vice-President, Secretary, and Treasurer be chosen
from 10 committee members?
Example: How many different 5-letter words can be formed from the letters in the word
“decagon”?
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COMBINATIONS – The number of ways that r items can be selected from n items (where order does
not matter) is:
 n
n!
nCr =   
 r  (n  r )! r!
(an arrangement of things in which the order makes no difference)
Example: How many ways can you pick 6 committee members from a group of 20 people?
Example: How many ways can a researcher select 5 rats from 20 rats and assign each a task?
Example: The Texas Lottery: Pick 6 numbers from 1 – 54.
a) How many number combinations are there?
b) What is the probability of winning the Texas Lottery?
Example: A company is setting up 4-digit ID numbers for employees.
a) How many 4-digit numbers are there if numbers can start with 0 and numbers can be repeated?
b) How many 4-digit numbers are there if numbers can start with 0 and the numbers must be
different?
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c) If you randomly assign an employee a 4-digit number that can start with 0 (and have repeats),
then what is the probability that all of the digits are even?
Example: Outside a home, there is a keypad that can be used to open the garage if the correct fourdigit code is entered.
a) How many codes are possible?
b) What is the probability that a burglar would enter the correct code on the first try?
Finding Probabilities Using “Categories”
Example: An animal shelter has 8 Dogs and 2 Cats up for adoption.
Suppose you randomly select 4 pets from the shelter.
Find the following probabilities:
a) P (3 Dogs, 1 Cat)
b) P (all 4 are Dogs)
Example: A sock drawer contains 14 white socks, 12 black socks, and 9 red socks. Three socks are
chosen from the drawer. What is the probability that all three socks are different colors?
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