Is the probability of getting a woman given the passenger was rescued equal to the probability of getting a rescued passenger given that they were a woman? Explain.

Question

https://imgur.com/a/OJZArjh Image with data tables here.

Answer 1

Whenever you see the word given, think restriction. Given restricts the sample space to only those outcomes that match the given criteria.

The probability of a woman given that that the passenger was rescued restricts up to the table of rescued passengers. There was a total of 499 rescued passengers (if we ignore the 212 rescued crew) of which 296 were women (if we ignore the 20 female crew members). The probability of a woman given they were rescued is 296/499 or 0.5932.

The probability of a rescued passenger given that they were a woman restricts the table down to the two columns that deal with women (both lost and rescued). There were a total of 296 rescued and 106 lost women for a grand total of 402 (This again ignores the 20 rescued and 3 lost female crew members). Since 296 of the 402 women were rescued, the probability is 296/402 or 0.7363.

As a general rule, reversing what is given in a conditional probability alters the probability enough that the two versions aren't the same.