[Math] Given the following life expectancy data, what is the probability of living to age 70


The problem: Let $P(n)$ be the probability of reaching the age of $n$ years. Suppose that we are given $P(50) = .913; P(55) = .881; P(65) = .746$. If the probability that a man who just turned $65$ will die within $5$ years is $.16$, what is the probability for a man to survive until his $70^{th}$ birthday, i.e., what is $P(70)$?

I think I know what the answer is, but my solution may as well be magic – I have little idea how or why my answer is correct. I know it has something to do with conditional probability but not much more than that. Can someone walk me through this problem? What are the concepts that I should be seeing?

(btw, I think the problem may be awkwardly worded – it's the professor's)

Best Answer

$P(70)=P($living 5 more years$) \cap P(65)=(1-.16)(.746)=.62664.$

EDIT: Since we know that the probability that the man will live $5$ more years if he is $65$ is $1-.16=.84$. This is equivalent to saying:


Recall that $P(A|B)=\frac{P(A\cap B)}{P(B)}$. So $P(70|65)=\frac{P(70\cap(65)}{P(65)}=\frac{P(70)}{P(65)}$

$\therefore P(70)=P(65)P(70|65)$, yielding the above result.

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