EXAMPLE 20 Determining whether two events are independent

Table 22 contains a contingency table summarizing the gender and survival status of the passengers and crew of RMS Titanic.

Table 5.55: Table 22 Contingency table of gender and survival status of passengers and crew of RMS Titanic. Notice that when we are given that the person is a female, we may restrict our attention to the females in the table.
Female Male Total
Did not survive 126 1364 1490
Survived 344 367 711
Total 470 1731 2201
Table 5.55: Source: Report on the Loss of the “Titanic” (S.S.). British Board of Trade Inquiry Report (reprint), Allan Sutton Publishing, Gloucester, United Kingdom, 1990.

Suppose our experiment is to select one person at random from the passengers and crew. Define the following events:

Determine whether events and are independent.

Solution

We use the strategy for determining whether two events are independent.

  • Step 1 Find . There are 470 females out of a total of 2201 passengers and crew. Thus, .
  • Step 2 We need to find , which is the probability that the person survived, given the person was female. We are told that the person was female, so we may restrict our attention to the females in the table. In other words, our sample space is reduced when we know that the person was a female. Of the 470 females, 344 survived, so that .
  • Step 3 Because, , we conclude that and are dependent events.

NOW YOU CAN DO

Exercise 33–52.