Actuarial Problem. (Policyholder).What is the probability that a new policyholder will have an accident within a year of purchasinag a policy?

Question

Problem said: Suppose people can be divided into two classes: those who are accident-prone and those who are not. The statistics show that an accident-prone person will have an accident at some time within a fide 1-year period with probability 0.35, whereas this probability for a non-accidenta-prone person is 0.18. Assume that 30.3% of the population is accident-prone.

What is the probability that a new policyholder will have an accident within a year of purchasinag a policy?

I apply the law of the total probability:

I have:

P(a new policyholder will have an accident )=(0.303)(.35)+(.697)(0.18)=0.23152

Is that correct?

Thanks.

Answer 1

This is a typical question that will be asked in Exam P as far as I can remember, however, I'd like to solve this type of question by the table method suggested by Weishaus in ASM EXAM C Lesson 45.

1.In the first row, enter the prior probability.

2.In the second row, enter the likelihood of experience given the class.

3.The third row is the product of the first two rows.

...

The full table would have 6 rows but in this case the first three are enough, so

                          accident-prone  nonaccident-prone
prior probability         0.303           0.697
likelihood of experience  0.35            0.18
joint probability         0.10605         0.12546

The total probability is 0.10605+0.12546=0.23151

Answer 2

The standard expression should be

$P(new\ holder\ has\ an\ accident)=P(new\ holder\ has\ an\ accident\ and\ he\ is\ acccident-prone )+P(new\ holder\ has\ an\ accident\ and\ he\ is\ accident-free)$

Then based on conditional probability, $P(new\ holder\ has\ an\ accident\ and\ he\ is\ risk-prone )=P(new\ holder\ has\ an\ accident\ | he\ is\ risk-prone)*P(he\ is\ risk-prone)= 0.35*0.303$

You can do the same calculation for the other part. and your calculation is correct. At all, it is called the law of total probability. May you refer to (https://en.wikipedia.org/wiki/Law_of_total_probability)