ProbabilityCh. 5

• General Multiplication Rule

The probability that events A and B both occur can be found using the general multiplication rule

P(A ∩ B) = P(A) • P(B | A)

where P(B | A) is the conditional probability that event B occurs given that event A has already occurred.

Playing in the NCAA

About 55% of high school students participate in a school athletic team at some level, and about 5% of these athletes go on to play on a college team in the NCAA

What percent of high school students play a sport in high school and go on to play a sport in the NCAA?

Late for school

Shannon hits the snooze bar on her alarm clock on 60% of school days. If she doesn’t hit the snooze bar, there is a 0.90 probability that she makes it to class on time. However, if she hits the snooze bar, there is only a 0.70 probability that she makes it to class on time. On a randomly chosen day, what is the probability that Shannon is late for class?

Suppose that Shannon is late for school. What is the probability that she hit the snooze bar that morning? We want to find

The Kaiser Family Foundation recently released a study about the influence of media in the lives of young people aged 8–18. In the study, 17% of the youth were classified as light media users, 62% were classified as moderate media users, and 21% were classified as heavy media users. Of the light users who responded, 74% described their grades as good (A’s and B’s), while only 68% of the moderate users and 52% of the heavy users described their grades as good. Suppose that we selected one young person at random.

(a) Draw a tree diagram to represent this situation. (b) Find the probability that this person describes his or her grades as good.(c) Given that this person describes his or her grades as good, what is the probability that he or she is a heavy user of media?

False positives and drug testing

Many employers require prospective employees to take a drug test. A positive result on this test indicates that the prospective employee uses illegal drugs. However, not all people who test positive actually use drugs. Suppose that 4% of prospective employees use drugs, the false positive rate is 5%, and the false negative rate is 10%.

A randomly selected prospective employee tests positive for drugs. What is the probability that he actually took drugs?

Conditional Probability and

IndependenceWhen knowledge that one event has happened does not change the likelihood that another event will happen, we say that the two events are independent.

Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if

P(A | B) = P(A) and P(B | A) = P(B).

When events A and B are independent, we can simplify the general multiplication rule since P(B| A) = P(B).

Multiplication rule for independent eventsIf A and B are independent events, then the probability that A and B both occur is

P(A ∩ B) = P(A) • P(B)

Is there a relationship between gender and relative finger length? To find out, we used the random sampler at the United States CensusAtSchool Web site to randomly select 452 U.S. high school students who completed a survey. The two-way table shows the gender of each student and which finger was longer on their left hand (index finger or ring finger).

Dominant Hand

Female Male Total

Index Finger

78 45 123

Ring Finger 82 152 234

Same Length

52 43 95

Total 212 240 452Are the events “female” and “has a longer ring finger” independent? Justify your answer.

In baseball, a perfect game is when a pitcher doesn’t allow any hitters to reach base in all nine innings. Historically, pitchers throw a perfect inning—an inning where no hitters reach base—about 40% of the time So, to throw a perfect game, a pitcher needs to have nine perfect innings in a row.

What is the probability that a pitcher throws nine perfect innings in a row, assuming the pitcher’s performance in an inning is independent of his performance in other innings?

With 30 teams playing 162 games per season, this means we would expect to see about (30)(162)(0.00026) = 1.3 perfect games per season. However, in baseball history, perfect games are much rarer, occurring once every 4 or 5 seasons, on average. This discrepancy suggests that our assumption of independence may not have been a good one.