In a city, the probability of men living to 90 years old is 0.7. The probability of men living to 100 years old is 0.5.

If a man has already reached 90 years old, what is the conditional probability that he will live to 100?


HINT: First, here’s an informal way to think about it. On average $7$ of every $10$ men reach the age of $90$, and $5$ of $10$ reach the age of $100$. In other words, out of every $7$ who reach $90$, on average $5$ reach $100$. What does this tell you about the probability of reaching $100$, given that a man has already reached $90$?

More formally, let $A$ be the event of reaching $90$ and $B$ the event of reaching $100$. You’re told that $\Bbb P(A)=0.7$ and $\Bbb P(B)=0.5$. You know that

$$\Bbb P(B\mid A)=\frac{\Bbb P(B\cap A)}{\Bbb P(A)}\;.$$

What is the event $B\cap A$? What is its probability?

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