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!


The Bayes Theorem states that:


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


That is what the Bayes Theorem actually says here.


First define your events, and use them to express what you have been told and seek in symbols.

Let $S$ be the event of a (specified) policy holder having a standard polity, and $D$ the event of that policy holder dying. You are given:

$$\begin{align}\mathsf P(S)&=0.60\\\mathsf P(D\mid S)&=0.01\\\mathsf P(S^\complement)&=0.49\\\mathsf P(D\mid S^\complement)&=0.08\end{align}$$

You seek $\mathsf P(S\mid D)$ and may now find it with the above using Bayes' Rule, and the Law of Total Probability.


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