The life insurance company issues standard and preferred policies.
Of all policyholders:
60% have standard policy and probability of 0.01 of dying next year.
40% have preferred policy and probabilty of 0.08 of dying next year.
A policyholder dies in the next year.
What is the conditional probability that a deceased having a preferred policy?
I know this could be solved via Bayes Theorem, but I am completely confused where to plug all the number at. Thank you!
Best Answer
The Bayes Theorem states that:
$P(A|B)=\frac{P(B|A)*P(A)}{P(B)}$
Now for your case let's define $A$ as the event of a person having a preferred policy.
And let's define $B$ as the event of a person dying next year.
From the Bayes Theorem we have:
$P(A|B)=\frac{P(B|A)*P(A)}{P(B)}=\frac{0.08*0.4}{0.08*0.4+0.01*0.6}=\frac{32}{32+6}\simeq 0.84$
Here is a way(not the most mathematical perhaps) to think of it if it confuses you:
You have 1000 people.
600 of them have standard policy. 400 have preferred policy. The ones that are going to die next year are 600*0.01 from the ones with standard policy and 400*0.08 from the ones with preferred policy. So a total of $600*0.01+400*0.08$ people are going to die next year. So given that someone died, the probability of him/her being from the ones with preferred policy is
$\frac{400*0.08}{600*0.01+400*0.08}$
That is what the Bayes Theorem actually says here.