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80 Introduction to probability
given that event B has occurred is normally written as p(A|B), so in our
case we wish to find: p(worker contracts cancer|exposed to chemical).
We only have 355 records of workers who were exposed to the chemical
and of these 220 have contracted cancer, so:
p(worker contracts cancer|exposed to chemical) = 220/355 = 0.620
Note that this conditional probability is greater than the marginal
probability of a worker contracting cancer (0.268), which implies that
exposure to the chemical increases a worker’s chances of developing
cancer. We will consider this sort of relationship between events next.
Independent and dependent events
Two events, A and B, are said to be independent if the probability of
event A occurring is unaffected by the occurrence or non-occurrence of
event B. For example, the probability of a randomly selected husband
belonging to blood group O will presumably be unaffected by the fact
that his wife is blood group O (unless like blood groups attract or repel!).
Similarly, the probability of very high temperatures occurring in England
next August will not be affected by whether or not planning permission
is granted next week for the construction of a new swimming pool at a
seaside resort. If two events, A and B, are independent then clearly:
p(A|B) = p(A)
because the fact that B has occurred does not change the probability of
A occurring. In other words, the conditional probability is the same as
the marginal probability.
In the previous section we saw that the probability of a worker
contracting cancer was affected by whether or not he or she has been
exposed to a chemical. These two events are therefore said to be dependent.
The multiplication rule
We saw earlier that the probability of either event A or B occurring can be
calculated by using the addition rule. In many circumstances, however,
we need to calculate the probability that both A and B will occur. For