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?
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.
(or indeed, if *everyone* survived)... If everyone survived then 100% of all women survived, regardless of the proportions.
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Commented
Jul 6, 2018 at 17:46
At its face, the conditional probability of surviving conditional on sex is more useful, simply because of the direction of information flow. A person's sex is known before her or his survival status, and this probability can be used in a predictive sense, prospectively. Also, it is not influenced by the prevalence of females. When in doubt, think prediction.
The first indicates that saving women was probably of high priority (irrespective of whether saving men was)
The word "priority" comes from the Latin for "before". A priority is something one comes before something else (where "before" is being used in the sense of "more important"). If you say that saving women was a priority, then saving women has to come before something else. And the natural assumption is that what it comes before is saving men. If you say "irrespective of whether saving men was", then we're left wondering what it came before.
That women had a high survival rate doesn't say much, if we don't know what the general survival rate was. The last ship I was on, over 90% of the women survived, but I wouldn't characterize that as showing that saving women was a high priority.
And knowing what percentage of survivors were women doesn't say much without knowing what percentage of people overall were women.
What statistic is more useful really depends on the situation. If you want to know how dangerous something is, the death rate is more important. If you want to know what affects how dangerous something is, then percentage breakdown of casualties is important.
It is possibly useful for us to examine how these probabilities are related.
Let $W$ be the event that a person is a woman, and let $S$ be the event that a person survived.
Statement 1: $$P(S|W) = 0.9$$
Statement 2: $$P(W|S) = 0.9$$
Bayes Theorem illustrates how these statements of probability are related.
$$P(S|W) = P(W|S)\frac{P(S)}{P(W)}$$
In this particular case, $P(S)$ (the probability of survival) and $P(W)$ (the proportion of Women on the titanic) are quite easy to look up, and therefore the probabilities are dependent on each other. That is, knowing one fully defines the other.
Treating $P(S)$ and $P(W)$ as known, they are the different ways of expressing the same information (albeit with different interpretations).
It depends on what what one considers useful.
If one is primarily interested in whether women were given higher priority than men, i.e. whether $P(S|W) > P(S|M)$, then both statements are equally useless without more information, as @StephanKolassa and @knrumsey have already said in their answers. If someone is meaning to express this kind of information, they'd need to say something more than statement 1, such as "90 percent of the women survived, but only 20 percent of the men survived".
On the other hand, if you're wondering why survivor stories are mostly from women, then statement 2 would explain that, making statement 2 useful even in the absence of other information.
I can't think of anything statement 1 is useful for out of context. It certainly doesn't say anything about the priority given to saving women, compared to anything else. The only thing statement 1 does for me is it makes me say "tell me more".
On the surface (or in isolation from reality) both statements appear to be equally useless for the state goal. However, considering the context, the second statement is clearly more useful.
Let's see what we can extract from the second statement. The ratio of women $w$ among all survived is: $$w = p x /(p x +(1-p) z) $$ where $p$ - ratio of women among passengers, $x$ and $z$ are probabilities of survival of women and men. The denominator is the total survival rate.
We are testing hypo $H_0:x>z$
Let's re-write the equation to obtain the necessary conditions for $H_0$:
$$(1-w) p x = w (1-p) z$$ $$ x = w (1-p) z/((1-w) p)$$ For $H_0$ to hold we have: $$ x = w (1-p) z/((1-w) p)>z$$ $$ w (1-p) >(1-w) p $$ $$ 0.9 (1-p) >0.1 p $$ $$ 1-p > p/9 $$ $$p<0.9$$
So, for your hypo that women were more likely to survive, all you need is to check that there were less than 90% women among the passengers. This is consistent with your assumption 2, which seems to imply that $p\approx 1/2$. Hence, I declare that statement 2 all but asserts that women were more likely to survive, i.e. it's quite useful for your goal.
The first statement is truly useless in isolation, but has a limited use in the context. If we pretend we know nothing about the event, then saying that $x=0.9$ tells us nothing about $z$, and whether $x>z$?
However, from that little that I know about the event - I haven't seen the movie - it seems unlikely that $x\le z$. Why?
We know from Assumption 2 that $p\approx 1/2$, so the total survival rate is $p x+(1-p) z$. If we assume that $x\approx z$ and $p\approx 1/2$ we get $$p x+(1-p) z\approx x=0.9$$ In other words 90% of all passengers survived, which doesn't ring true to me. Would they make a movie and talk about it for 100 years if 90% of passengers survived? So, it must be that $x>>z$ and less than half of passengers made it.
I'd say that both statements support your hypo that women were more likely to survive than men, but Statement 1 does so rather weakly, while Statement 2 in combination with assumptions almost surely establishes your hypo as a fact.
