1. 6.5 Combining Probabilities (Supplementary Section)
LEARNING GOAL
Distinguish between independent and dependent events and between overlapping and non-overlapping events, and be able to calculate and and either/or probabilities.
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And Probabilities
Suppose you toss two fair dice and want to know the probability that both will come up 4.
For each die, the probability of a 4 is 1/6. We find the probability that both dice show a 4 by multiplying the individual probabilities:
P(double 4’s) = P(4) × P(4) = × =
In general, we call the probability of event A and event B occurring an and probability (or joint probability). 1 6 36
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3. And Probabilities Independent Events
And Probability for Independent Events
Two events are independent if the outcome of one event does not affect the probability of the other event. Consider two independent events A and B with probabilities P(A) and P(B). The probability that A and B occur together is
P(A and B) = P(A) × P(B)
This principle can be extended to any number of independent events. For example, the probability of A, B, and a third independent event C is
P(A and B and C) = P(A) × P(B) × P(C)
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EXAMPLE 1 Three Coins
EXAMPLE 1 Three Coins
Suppose you toss three fair coins. What is the probability of getting three tails?
Solution: Because coin tosses are independent, we multiply the probability of tails on each individual coin:
P(3 tails) = P(tails) × P(tails) × P(tails) = × × =
The probability that three tossed coins all land on tails is 1/8 (which we determined in Example 8 of Section 6.2 with much more work). 1 2 1 2 1 2 1 8
coin coin coin 3
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And Probabilities
Independent Events
Dependent Events
A batch of 15 memory cards contains 5 defective cards. What is the probability of getting a defect card on both the first and the second selection?
P(2 defectives) = P(defective) × P(defective)
= × =
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6. And Probability for Dependent Events
Two events are dependent if the outcome of one event affects the probability of the other event. The probability that dependent events A and B occur together is
P(A and B) = P(A) × P(B given A)
where P(B given A) means the probability of event B given the occurrence of event A.
This principle can be extended to any number of individual events. For example, the probability of dependent events A, B, and C is
P(A and B and C) = P(A) × P(B given A)
× P(C given A and B)
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TECHNICAL NOTE
P(B given A) is called a conditional probability. In some books, it is denoted P(B|A).
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TIME OUT TO THINK
Without doing calculations, compare the probability of drawing three hearts in a row when the card is replaced and the deck is shuffled after each draw to the probability of drawing three hearts in a row without replacement. Which probability is larger and why?
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EXAMPLE 3 When Can We Treat Dependent
Events as Independent Events?
A polling organization has a list of 1,000 people for a telephone survey. The pollsters know that 433 people out of the 1,000 are members of the Democratic Party. Assuming that a person cannot be called more than once, what is the probability that the first two people called will be members of the Democratic Party?
Solution: This problem involves an and probability for dependent events: Once a person is called, that person cannot be called again. The probability of calling a member of the Democratic Party on the first call is 433/1,000.
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EXAMPLE 3 When Can We Treat Dependent
Events as Independent Events?
Solution: (cont.)
With that person removed from the calling pool, the probability of calling a member of the Democratic Party on the second call is 432/999. Therefore, the probability of calling two members of the Democratic Party on the first two calls is
433
1,000
432
999
× =
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EXAMPLE 3 When Can We Treat Dependent
Events as Independent Events?
Solution: (cont.)
If we treat the two calls as independent events, the probability of calling a member of the Democratic Party is 433/1,000 on both calls. Then the probability of calling two members of the Democratic Party is
which is nearly identical to the result assuming dependent events (previous slide).
433
1,000
× =
1000
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EXAMPLE 3 When Can We Treat Dependent
Events as Independent Events?
Solution: (cont.)
In general, if relatively few items or people are selected from a large pool (in this case, 2 people out of 1,000), then dependent events can be treated as independent events with very little error.
A common guideline is that independence can be assumed when the sample size is less than 5% of the population size. This practice is commonly used by polling organizations.
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13. Either/Or Probabilities
Non-Overlapping Events
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Figure 6.14 Venn diagram for non-overlapping events.
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14. Either/Or Probability for Non-Overlapping Events
Two events are non-overlapping if they cannot occur at the same time. If A and B are non-overlapping events, the probability that either A or B occurs is
P(A or B) = P(A) + P(B)
This principle can be extended to any number of non-overlapping events. For example, the probability that either event A, event B, or event C occurs is
P(A or B or C) = P(A) + P(B) + P(C)
provided that A, B, and C are all non-overlapping events.
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EXAMPLE 5 Either/Or Dice Probability
Suppose you roll a single die. What is the probability of getting an even number?
Solution: The even outcomes of 2, 4, and 6 are non-overlapping because a single die can yield only one result. We know that P(2) = 1/6, P(4) = 1/6, and P(6) = 1/6. Therefore, the combined probability is
P(2 or 4 or 6) = P(2) + P(4) + P(6) =
The probability of rolling an even number is 1/2.
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Either/Or Probabilities
Non-Overlapping Events
Overlapping Events
To improve tourism between France and the United States, the two governments form a committee consisting of 20 people: 2 American men, 4 French men, 6 American women, and 8 French women (as shown in
Table 6.9). If you meet one
of these people at random,
what is the probability that
the person will be either a
woman or a French person?
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Twelve of the 20 people are French, so the probability of meeting a French person is 12/20.
Similarly, 14 of the 20
people are women, so
the probability of
meeting a woman is
14/20.
However the eight
French women were
included in both
groups.
Thus
Pages 14 20 12 20 8 20 18 20 9 10
P(woman or French) = – = =
probability
of a woman
probability of
a French person
probability of
a French woman
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18. Either/Or Probability for Overlapping Events
We say that meeting a woman and meeting a French person are overlapping (or non–mutually exclusive) events because both can occur at the same time.
Either/Or Probability for Overlapping Events
Two events A and B are overlapping if they can occur together. The probability that either A or B occurs is
P(A or B) = P(A) + P(B) – P(A and B)
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The last term, P(A and B), corrects for the double counting of events in which A and B both occur together.
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TIME OUT TO THINK
Are the events of being born on a Wednesday or being born in Las Vegas overlapping? Are the events of being born on a Wednesday or being born in March overlapping? Are the events of being born on a Wednesday or being born on a Friday overlapping? Explain.
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Summary
Table 6.11 provides a summary of the formulas we’ve used in combining probabilities.
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The End
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