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)
 A: $P(70)=P($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:
$P(70|65)=.84$ 
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.
A: In general when you're given the probabilities of several related events, and then asked to compute the probability of some other related event, you should try to partition or express the new event in terms of the previously mentioned ones.
For example, if one would like the probability of a given person getting married in the next year, and one knows the probability of a male getting married in the next year, and the probability of a female getting married in the next year, one has partitioned the event "a person getting married in the next year" into two mutually exclusive ones. This allows you to add probabilities. 
In this case it's even easier. You should try to think to yourself what MUST be true if a man survives until his 70th birthday.....
Hint: He has to survive until his 65th, and then additionally live at least a certain number of years....
