as is the probability that a flier would both contain a message about littering and be found
stuffed down behind the Raisin Bran. Given two events, their joint probability is denoted as
p(A, B), just as we have used p(blue, green) or p(message, trash). If those two events are in-
dependent, then the probability of their joint occurrence can be found by using the multi-
plicative law, as we have just seen. If they are notindependent, the probability of their joint
occurrence is more complicated to compute and will differ from what it would be if the
events were independent. We won’t compute that probability here.
A conditional probabilityis the probability that one event will occur giventhat some
other event has occurred. The probability that a person will contract AIDS given that he or
she is an intravenous drug user is a conditional probability. The probability that an adver-
tising flier will be thrown in the trash given that it contains a message about littering is an-
other example. A third example is a phrase that occurs repeatedly throughout this book: “If
the null hypothesis is true, the probability of obtaining a result such as this is....” Here I
have substituted the word iffor given, but the meaning is the same.
With two events, A and B, the conditional probability of A given B is denoted by use of
a vertical bar, as p(A | B), for example, p(AIDS | drug user) or p(trash | message).
We often assume, with some justification, that parenthood breeds responsibility. Peo-
ple who have spent years acting in careless and irrational ways somehow seem to turn into
different people once they become parents, changing many of their old behavior patterns.
(Just wait a few years.) Suppose that a radio station sampled 100 people, 20 of whom had
children. They found that 30 of the people sampled used seat belts, and that 15 of those
people had children. The results are shown in Table 5.2.
The information in Table 5.2 allows us to calculate the simple, joint, and conditional
probabilities. The simple probability that a person sampled at random will use a seat belt is
30/100 5 .30. The joint probability that a person will have children andwill wear a seat belt
is 15/100 5 .15. The conditional probability of a person using a seat belt given that he or she
has children is 15/20 5 .75. Do not confuse joint and conditional probabilities. As you can
see, they are quite different. You might wonder why I didn’t calculate the joint probability
here by multiplying the appropriate simple probabilities. The use of the multiplicative law re-
quires that parenthood and seat belt use be independent. In this example they are not, because
the data show that whether people use seat belts depends very much on whether or not they
have children. (If I had assumed independence, I would have predicted the joint probability
to be .30 3 .20 5 .06, which is less than half the size of the actual obtained value.)
To take another example, the probability that you have been drinking alcoholic beverages
and that you have an accident is a joint probability. This probability is not very high, because
relatively few people are drinking at any one time and relatively few people have accidents.
However, the probability that you have an accident given that you have been drinking, or, in
reverse, the probability that you have been drinking given that you have an accident, are both
much higher. At night the conditional probability of p(drinking | accident) approaches .50,
since nearly half of all automobile accidents at night in the United States involve alcohol. I
don’t know the conditional probability of p(accident | drinking), but I do know that it is much
higher than the unconditional probabilityof an accident, that is, p(accident).Section 5.2 Basic Terminology and Rules 117Table 5.2 The relationship between parenthood and seat belt use
Wear Seat Do Not Wear
Parenthood belt Seat belt Total
Children 15 5 20
No children 15 65 80
Total 30 70 100conditional
probability
unconditional
probability