The more important statistic: ’90 percent of all women survived’ or ’90 percent of all those who survived were women’?

Consider the following statements w.r.t the Titanic:

Assumption 1: Only men and women were on the ship

Assumption 2: There were a large number of men as well as women

Statement 1: 90 percent of all women survived

Statement 2: 90 percent of all those who survived, were women

The first indicates that saving women was probably of high priority (irrespective of whether saving men was)

When is the second statistic useful?

Can we say that one of them is almost always more useful than the other?

Answer

As they stand, neither one of Statement 1 or 2 is very useful. If 90% of passengers were women and 90% of people survived at random, then both statements would be true. The statements need to be considered in the context of the overall composition of the passengers. And the overall chance of surviving.


Suppose we had as many men as women, 100 each. Here are a few possible matrices of men (M) against women (W) and surviving (S) against dead (D):

  |  M |  W
------------
S | 90 | 90
------------
D | 10 | 10

90% of women survived. As did 90% of men. Statement 1 is true, Statement 2 is false, since half of survivors were women. This is consistent with many survivors, but no difference between genders.

  |  M |  W
------------
S | 10 | 90
------------
D | 90 | 10

90% of women survived, but only 10% of men. 90% of the survivors were women. Both statements are true. This is consistent with a difference between genders: women were more likely to survive than men.

  |  M |  W
------------
S |  1 |  9
------------
D | 99 | 91

9% of women survived, but only 1% of men. 90% of the survivors were women. Statement 1 is false, Statement 2 is true. This is again consistent with a difference between genders: women were more likely to survive than men.

Attribution
Source : Link , Question Author : rahs , Answer Author : Stephan Kolassa

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