Download PROBABILITY

14.3
Probability
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14.3 P R O B A B I L I T Y
In this
section
●
The Probability of an Event
●
The Addition Rule
●
Complementary Events
●
Odds
In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques to find
probabilities.
The Probability of an Event
In probability an experiment is a process such as tossing a coin, tossing a die,
drawing a poker hand from a deck, or arranging people in a line. A sample space is
the set of all possible outcomes to an experiment. An event is a subset of a sample
space. For example, if we toss a coin, then the sample space consists of two equally
likely outcomes, heads and tails. We write S H, T. The subset E H is the
event of getting heads when the coin is tossed. We use n(S) to represent the number
of equally likely outcomes in the sample space S and n(E ) to represent the number
of outcomes in the event E. For the example of tossing a coin, n(S ) 2 and
n(E ) 1.
The Probability of an Event
If S is a sample space of equally likely outcomes to an experiment and the
event E is a subset of S, then the probability of E, P(E), is defined to be
n(E)
P(E) .
n(S)
When S H, T and E H,
n(E) 1
P(E) .
n(S)
2
1
So the probability of getting heads on a single toss of a coin is 2. If E is the event of
0
getting 2 heads on a single toss of a coin, then n(E) 0 and P(E) 2 0. If E
is the event of getting fewer than 2 heads on a single toss of a coin, then for either
outcome H or T we have fewer than 2 heads. So E H, T, n(E) 2, and
2
P(E) 2 1. Note that the probability of an event is a number between 0 and 1
inclusive, 1 being the probability of an event that is certain to occur and 0 being the
probability of an event that is impossible to occur.
E X A M P L E
1
Rolling a die
What is the probability of getting a number larger than 4 when a single die is rolled?
Solution
When we roll a die, we count the number of dots showing on the upper face of
the die. So the sample space of equally likely outcomes is S 1, 2, 3, 4, 5, 6.
Since only 5 and 6 are larger than 4, E 5, 6. According to the definition of
probability,
n(E ) 2 1
P(E ) .
n(S)
6 3
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Chapter 14
Counting and Probability
M A T H
A T
W O R K
The probability experiments discussed in this chapter are not just
textbook examples that have no relationship to real life. For example, if a
couple plans to have 6 children and
LOTTERIES
the probability of having a girl on
1
each try is 2, then the couple can expect to have 3 girls. If you guess at the answer
1
to each question of a 100-question, 5-choice multiple-choice test, then you have 5
probability of getting each question correct, and you can expect to get 20 questions
correct. Try it. The expected number of successes is the product of the probability
of success and the number of tries.
Lotteries provide us an opportunity to observe massive probability experiments.
In the Florida Lottery you can win by choosing 6 numbers from the numbers 1
through 49 and matching the 6 numbers chosen by the Florida Lottery. There are
C(49, 6) ways to choose 6 numbers from 49, so the probability of winning on any
individual try is
1
1
.
C(49, 6) 13,983,816
In the fall of 1990 the weekly drawing frequently had relatively few participants,
and consequently there was no winner for many weeks. When the prize got up to
$106.5 million, the lottery got national attention. People came from everywhere to
participate. During the week prior to September 15, 1990, 109,163,978 tickets were
1
sold. We expected 109,163,978 7.8 winners. On September 15 the
13,983,816
winning numbers were announced, and 6 winners shared the prize. Of course, probability cannot predict the future like a fortune-teller, but the power of probability to
tell us what to expect is truly amazing.
E X A M P L E
2
Tossing coins
What is the probability of getting at least one head when a pair of coins is tossed?
Solution
Since there are 2 equally likely outcomes for the first coin and 2 equally likely
outcomes for the second coin, by the fundamental counting principle there are 4
equally likely outcomes to the experiment of tossing a pair of coins. We can list the
outcomes as ordered pairs: S (H, H), (H, T), (T, H), (T, T). Since 3 of these
outcomes result in at least one head, E ( H, H), (H, T ), (T, H), and n(E) 3.
So
n(E) 3
P(E) .
n(S)
4
E X A M P L E
3
Rolling a pair of dice
What is the probability of getting a sum of 6 when a pair of dice is rolled?
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14.3
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Probability
729
Solution
Since there are 6 equally likely outcomes for each die, there are 36 equally likely
outcomes to the experiment of rolling the pair. We can list the 36 outcomes as
ordered pairs:
S (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
The sum of the numbers is 6, describes the event E (5, 1), (4, 2), (3, 3), (2, 4),
(1, 5). So
5
n(E)
P(E) .
n(S)
36
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The Addition Rule
In tossing a pair of dice, let A be the event that doubles occurs and B be the event
that the sum is 4. We can list the following events and their probabilities:
6
A (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) and P(A) 36
3
B (3, 1), (2, 2), (1, 3) and P(B) 36
A B (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (3, 1), (1, 3)
8
and P(A B) 36
1
A B (2, 2) and P(A B) 36
8
Note that the probability of doubles or a sum of 4, P(A B), is 36 and
8
6
3
1
.
36 36 36 36
This equation makes sense because there is one outcome, (2, 2), that is in both the
events A and B. This example illustrates the addition rule.
The Addition Rule
If A and B are any events in a sample space, then
P(A B) P(A) P(B) P(A B).
If P(A B) 0, then A and B are called mutually exclusive events and
P(A B) P(A) P(B).
Note that for mutually exclusive events it is impossible for both events to occur.
The addition rule for mutually exclusive events is a special case of the general
addition rule.
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Chapter 14
E X A M P L E
4
Counting and Probability
The addition rule
At Downtown College 60% of the students are commuters (C), 50% are female (F),
and 30% are female commuters. If a student is selected at random, what is the
probability that the student is either a female or a commuter?
Solution
By the addition rule the probability of selecting either a female or a commuter is
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P(F C) P(F) P(C) P(F C) 0.50 0.60 0.30 0.80.
E X A M P L E
5
The addition rule with dice
In rolling a pair of dice, what is the probability that the sum is 12 or at least one die
shows a 2?
Solution
Let A be the event that the sum is 12 and B be the event that at least one die shows
a 2. Since A occurs on only one of the 36 equally likely outcomes (see Example 3),
1
11
P(A) 36 . Since B occurs on 11 of the equally likely outcomes, P(B) 36 . Since
A and B are mutually exclusive, we have
1
11 12 1
P(A B) P(A) P(B) .
36 36 36 3
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Complementary Events
If the probability of rain today is 60%, then the probability that it does not rain is
40%. Rain and not rain are called complementary events. There is no possibility that
both occur, and one of them must occur. If A is an event, then A (read “A bar” or “A
complement”) represents the complement of the event A. Note that complementary
events are mutually exclusive, but mutually exclusive events are not necessarily
complementary.
Complementary Events
Two events A and A
are called complementary events if A A
and
) 1.
P(A) P(A
E X A M P L E
6
Complementary events
What is the probability of getting a number less than or equal to 4 when rolling a
single die?
Solution
We saw in Example 1 that getting a number larger than 4 when rolling a single die
1
has probability 3. The complement to getting a number larger than 4 is getting a
number less than or equal to 4. So the probability of getting a number less than or
2
equal to 4 is 3.
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E X A M P L E
7
Complementary events
If the probability that White Lightning will win the Kentucky Derby is 0.15, then
what is the probability that White Lightning does not win the Kentucky Derby?
14.3
study
tip
Study for the final exam by
working actual test questions.
Be sure to rework all of your
tests. Do the chapter tests in
this book. You can get more
tests to work by asking students or instructors for tests
that were given in other
classes of this course.
Probability
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Solution
Let W be winning the Kentucky Derby and N be not winning the Kentucky
Derby. Since W and N are complementary events, we have P(W) P(N) 1.
■
So P(N) 1 P(W) 1 0.15 0.85.
Odds
2
1
If the probability is 3 that the Giants win the Super Bowl and 3 that they lose, then
they are twice as likely to win as they are to lose. We say that the odds in favor of
the Giants winning the Super Bowl are 2 to 1. Notice that odds are not probabilities.
Odds are ratios of probabilities. We usually write odds as ratios of whole numbers.
Odds
If A is any event, then the odds in favor of A is the ratio P(A) to P(A
) and the
) to P(A).
odds against A is the ratio of P(A
E X A M P L E
8
Determining odds
What are the odds in favor of getting a sum of 6 when rolling a pair of dice? What
are the odds against a sum of 6?
Solution
5
In Example 3 we found the probability of a sum of 6 to be 36. So the probability of
31
the complement (the sum is not 6) is 36. The odds in favor of getting a sum of 6 are
5
31
to . Multiply each fraction by 36 to get the odds 5 to 31. The odds against a sum
36
36
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of 6 are 31 to 5.
E X A M P L E
helpful
9
hint
Odds and probability are
often confused, even by people who write lottery tickets. If
the probability of winning a
lottery is 1, then the proba-
Determining probability given the odds
If the odds in favor of Daddy’s Darling winning the third race at Delta Downs are 4
to 1, then what is the probability that Daddy’s Darling wins the third race?
Solution
Since 4 to 1 is the ratio of the probability of winning to not winning, the probability
) x
of winning is four times as large as the probability of not winning. Let P(W
) 1, we have 4x x 1, or 5x 1, or
and P(W ) 4x. Since P(W ) P(W
1
4
x 5. So the probability of winning is 5.
■
We can write the idea found in Example 9 as a strategy for converting from odds
to probabilities.
100
9
bility of losing is 9
, and the
100
odds in favor of winning are 1
to 99. Many lottery tickets will
state (incorrectly) that the
odds in favor of winning are 1
to 100.
Strategy for Converting from Odds to Probability
If the odds in favor of event E are a to b, then
a
P(E) ab
and
b
P(E
) .
ab
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Chapter 14
WARM-UPS
Counting and Probability
True or false? Explain your answer.
1. If S is a sample space of equally likely outcomes and E is a subset of S, then
P(E) n(E ). False
2. If an experiment consists of tossing 3 coins, then the sample space consists
of 6 equally likely outcomes. False
3. The probability of getting at least one tail when a coin is tossed twice is 0.75.
True
11
4. The probability of getting at least one 4 when a pair of dice is tossed is 36.
True
33
5. The probability of getting at least one head when 5 coins are tossed is 32.
False
6. If 3 coins are tossed, then getting exactly 3 heads and getting exactly 3 tails
are complementary events. False
1
7. If the probability of getting exactly 3 tails in a toss of 3 coins is 8, then the
7
probability of getting at least one head is 8. True
8. If the probability of snow today is 80%, then the odds in favor of snow are
8 to 10. False
2
9. If the odds in favor of an event E are 2 to 3, then P(E) 3. False
1
1
10. The ratio of 2 to 3 is equivalent to the ratio of 2 to 3. False
14. 3
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is an experiment?
An experiment is a process for which the outcomes are
uncertain.
2. What is a sample space?
A sample space is the set of all possible outcomes to an
experiment.
3. What is an event?
An event is a subset of a sample space.
4. What is the probability of an event?
The probability of an event is the ratio of the number of
outcomes in the event to the number of outcomes in the
sample space.
5. What is the addition rule?
The addition rule states that if A and B are events in a sample
space, then P(A B) P(A) P(B) P(A B).
6. What are the odds in favor of an event?
The odds in favor of an event is the ratio of the probability
of the event to the probability of the complement of the
event.
Solve each probability problem. See Example 1–3.
7. If a single die is tossed, then what is the probability of
getting
a) a number larger than 3?
b) a number less than or equal to 5?
c) a number other than 6?
d) a number larger than 7?
e) a number smaller than 9?
1 5 5
, , , 0, 1
2 6 6
8. If a single coin is tossed once, then what is the probability
of getting
a) tails?
c) exactly three heads?
b) fewer than two heads?
1
, 1, 0
2
9. If a pair of coins is tossed, then what is the probability of
getting
a) exactly two heads?
c) exactly two tails?
b) at least one tail?
d) at most one tail?
1 3 1 3
, , , 4 4 4 4
14.3
10. If a single coin is tossed twice, then what is the probability
of getting
a) heads followed by tails? c) a tail on the second toss?
b) two heads in a row?
d) exactly one tail?
1 1 1 1
, , , 4 4 2 2
11. If a pair of dice is tossed, then what is the probability of
getting
a) a pair of 2’s?
d) a sum greater than 1?
b) at least one 2?
e) a sum less than 2?
c) a sum of 7?
1 11 6
, , , 1, 0
36 36 36
12. If a single die is tossed twice, then what is the probability of
getting
a) a 1 followed by a 2? 1
36
b) a sum of 3?
1
18
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Probability
b) the sum of the numbers selected is 3?
c) the sum of the numbers selected is 6?
733
1
45
2
45
16. A small company consists of a president, a vice-president,
and 14 salespeople. If 2 of the 16 people are randomly selected to win a Hawaiian vacation, then what is the probability that none of the salespeople is a winner? 1
120
17. If a 5-card poker hand is drawn from a deck of 52, then
what is the probability that
a) the hand contains the ace, king, queen, jack, and ten of
1
spades?
2,598,960
b) the hand contains one 2, one 3, one 4, one 5, and one 6?
1024
2,598,960
18. If 5 people with different names and different weights
randomly line up to buy concert tickets, then what is the
probability that
a) they line up in alphabetical order? 1
120
c) a 6 on the second toss? 1
6
d) no more than two 5’s? 1
e) an even number followed by an odd number?
b) they line up in order of increasing weight?
1
4
13. A ball is selected at random from a jar containing 3 red
balls, 4 yellow balls, and 5 green balls. What is the probability that
a) the ball is red? 1
4
b) the ball is not yellow?
2
3
c) the ball is either red or green?
d) the ball is neither red nor green?
2
3
1
3
1
120
Use the addition rule to solve each problem. See Examples 4
and 5.
19. Among the drivers insured by American Insurance, 65%
are women, 38% of the drivers are in a high-risk category,
and 24% of the drivers are high-risk women. If a driver is
randomly selected from that company, what is the probability that the driver is either high-risk or a woman? 0.79
20. What is the probability of getting either a sum of 7 or at
least one 4 in the toss of a pair of dice? 5
12
21. A couple plans to have 3 children. Assuming males and females are equally likely, what is the probability that they
have either 3 boys or 3 girls. 1
4
22. What is the probability of getting a sum of 10 or a sum of 5
in the toss of a pair of dice? 7
36
23. What is the probability of getting either a heart or an ace
when drawing a single card from a deck of 52 cards? 4
13
24. What is the probability of getting either a heart or a spade
when drawing a single card from a deck of 52 cards? 1
2
FIGURE FOR EXERCISE 13
14. A committee consists of 1 Democrat, 5 Republicans, and 6
independents. If one person is randomly selected from the
committee to be the chairperson, then what is the probability that
a) the person is a Democrat? 1
12
b) the person is either a Democrat or a Republican?
c) the person is not a Republican?
1
2
7
12
15. A jar contains 10 balls numbered 1 through 10. Two balls
are randomly selected one at a time without replacement.
What is the probability that
a) 1 is selected first and 2 is selected second? 1
90
Solve each problem. See Examples 6 and 7.
25. If the probability of surviving a head-on car accident
at 55 mph is 0.005, then what is the probability of not
surviving? 0.995
26. If the probability of a tax return not being audited by the
IRS is 0.97, then what is the probability of a tax return
being audited? 0.03
27. A pair of dice is tossed. What is the probability of
a) getting a pair of 4’s? 1
36
b) not getting a pair of 4’s?
35
36
c) getting at least one number that is not a 4? 35
36
28. Three coins are tossed. What is the probability of
a) getting three heads? 1
8
b) not getting three heads?
c) getting at least one tail?
7
8
7
8
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Chapter 14
Counting and Probability
Solve each problem. See Examples 8 and 9.
29. If the probability is 60% that the eye of Hurricane Edna
comes ashore within 30 miles of Charleston, then what are
the odds in favor of the eye of Edna coming ashore within
30 miles of Charleston? 3 to 2
FIGURE FOR EXERCISE 38
39. If one million lottery tickets are sold and only one of them
is the winning ticket, then what are the odds in favor of
winning if you hold a single ticket? 1 to 999,999
40. What are the odds in favor of winning a lottery where you
must choose 6 numbers from the numbers 1 through 49?
1 to 13,983,815
FIGURE FOR EXERCISE 29
30. If the probability that a Sidewinder missile hits its target is
8
,
9
then what are the odds
a) in favor of the Sidewinder hitting its target? 8 to 1
b) against the Sidewinder hitting its target? 1 to 8
31. If the probability that the stock market goes up tomorrow is
3
,
5
then what are the odds
a) in favor of the stock market going up tomorrow? 3 to 2
b) against the stock market going up tomorrow? 2 to 3
9
32. If the probability of a coal miners’ strike this year is 10, then
what are the odds
a) in favor of a strike? 9 to 1
b) against a strike? 1 to 9
33. If the odds are 3 to 1 in favor of the Black Hawks winning
their next game, then
a) what are the odds against the Black Hawks winning
their next game? 1 to 3
b) what is the probability that the Black Hawks win their
next game? 3
4
34. If the odds are 5 to 1 against the Democratic presidential
nominee winning the election, then
a) what are the odds in favor of the Democrat winning the
election? 1 to 5
b) what is the probability that the Democrat wins the election? 1
6
35. What are the odds in favor of getting exactly 2 heads in 3
tosses of a coin? 3 to 5
36. What are the odds in favor of getting a 6 in a single toss of
a die? 1 to 5
37. What are the odds in favor of getting a sum of 8 when tossing a pair of dice? 5 to 31
38. What are the odds in favor of getting at least one 6 when
tossing a pair of dice? 11 to 25
41. If the odds in favor of getting 5 heads in 5 tosses of a coin
are 1 to 31, then what is the probability of getting 5 heads
in 5 tosses of a coin? 1
32
42. If the odds against Smith winning the election are 2 to 5,
then what is the probability that Smith wins the election?
5
7
GET TING MORE INVOLVED
43. In the Louisiana Lottery a player chooses 6 numbers from
the numbers 1 through 44. You win the big prize if the 6
chosen numbers match the 6 winning numbers chosen on
Saturday night.
a) What is the probability that you choose all 6 winning
numbers?
b) What is the probability that you do not get all 6 winning
numbers?
c) What are the odds in favor of winning the big prize with
a single entry?
1
7,059,051
, , 1 to 7,059,051
7,059,052 7,059,052
44. In the Louisiana Power Ball a player chooses 5 numbers
from the numbers 1 through 49 and one number (the power
ball) from 1 through 42.
a) How many ways are there to choose the 5 numbers and,
choose the power ball?
b) What is the probability of winning the big prize in the
Power Ball Lottery?
c) What are the odds in favor of winning the big prize?
1
80,089,128, , 1 to 80,089,127
80,089,128