Question

You are the Claims Manager for a large auto insurance firm. You know the following facts: • 25% of your policy-holders are younger (age 16-34). Within this group, 55% are men and 45% are women. Only some of these policy holders make an accident claim. The men have a 10% probability of making a claim, while the women have a 5% probability of making a claim. • 50% of your policy-holders are middle-aged (ages 35-59). Of these, 50% are men and 50% are women. Only some of these policy holders make an accident claim. The men and women within this age group have 4% and 3% probabilities of making a claim, respectively. • 25% of your policy-holders are older (ages 60 and above). Of these, 55% are women and 45% are men. Only some of these policy holders make an accident claim. The men and women each have a 2% probability of making a claim. (a) What is the probability that a policy-holder who makes a claim is in the “16-34” age group? (b) What is the probability that a policy-holder who does not make a claim is in the “60 and above” age category?

Answer 1

Here 25% of policy-holders are (16-34) age group, 50% are in (35-59) age group , and 25% are in (60 and above).

Now, let total number of policy-holders among all age groups be 'x'.

Therefore.

Total number of policy-holders in 16-34 age group will be 0.25x ....( Since, 25% are in that age group)

Total number of policy-holders in 35-59 age group will be 0.50x ....( Since, 50% are in that age group)

Total number of policy-holders in 60 and above age group will be 0.25x ....( Since, 25% are in that age group)

(a)

in 16-34 age group 55% are men and 45% are women.

No. of men in the age group = 0.55 * 0.25x

No. of women in the age group = 0.45 * 0.25x

Men have a probability of making a 10% claim ,while women have a probability of making 5% claim.

No. of men making claims = 0.10 * 0.55 * 0.25x

No. of women making claims = 0.05 * 0.45 * 0.25x

Total number of people making a policy-holders claim =0.10 * 0.55 * 0.25x + 0.05 * 0.45 * 0.25x .......(1)

in 35-59 age group 50% are men and 50% are women.

No. of men in the age group = 0.50 * 0.50x

No. of women in the age group = 0.50 * 0.50x

Men have a probability of making a 4% claim ,while women have a probability of making 3% claim.

No. of men making claims = 0.04 * 0.50 * 0.50x

No. of women making claims = 0.03 * 0.50 * 0.50x

Total number of people making a policy-holders claim =0.04 * 0.50 * 0.50x + 0.03 * 0.50 * 0.50x .......(2)

in 60 and above age group 55% are men and 45% are women.

No. of men in the age group = 0.55 * 0.25x

No. of women in the age group = 0.45 * 0.25x

Men have a probability of making a 2% claim ,while women have a probability of making 2% claim.

No. of men making claims = 0.02 * 0.55 * 0.25x

No. of women making claims = 0.02 * 0.45 * 0.25x

Total number of people making a policy-holders claim =0.02 * 0.55 * 0.25x + 0.02 * 0.45 * 0.25x .....(3)

We have to find out the probability of claims in 16-34 age group

Therefore , probabiity= no. of people making claims in 16-34 age group / total no. of people making claims in all age group.

From (1) ,(2) and (3)

probability = 0.10 * 0.55 * 0.25x + 0.05 * 0.45 * 0.25x /  0.10 * 0.55 * 0.25x + 0.05 * 0.45 * 0.25x + 0.04 * 0.50 * 0.50x + 0.03 * 0.50 * 0.50x + 0.02 * 0.55 * 0.25x + 0.02 * 0.45 * 0.25x

probability = 0.01375 x + 0.005625x / 0.01375 x + 0.005625x + 0.01x + 0.0075x + 0.00275x + 0.00225x

probability =0.019375 / 0.019375 + 0.0175 + 0.005

probability =0.019375 / 0.041875

probability = 0.4626

Hence, probability that no. of people making claim is in the agegroup of 16-34 is 0.46

(b)

in 16-34 age group 55% are men and 45% are women.

No. of men in the age group = 0.55 * 0.25x

No. of women in the age group = 0.45 * 0.25x

Men not making a claim is 90% ,while women not making a claim 95% .

No. of men not making claims = 0.90 * 0.55 * 0.25x

No. of women not making claims = 0.95 * 0.45 * 0.25x

Total number of people not making a policy-holders claim =0.90 * 0.55 * 0.25x + 0.95 * 0.45 * 0.25x .......(4)

in 35-59 age group 50% are men and 50% are women.

No. of men in the age group = 0.50 * 0.50x

No. of women in the age group = 0.50 * 0.50x

Men not making a 96% claim ,while women not making a claim is 97%

No. of men not making claims = 0.96 * 0.50 * 0.50x

No. of women not making claims = 0.97 * 0.50 * 0.50x

Total number of people not making a policy-holders claim =0.96 * 0.50 * 0.50x + 0.97 * 0.50 * 0.50x .......(5)

in 60 and above age group 55% are men and 45% are women.

No. of men in the age group = 0.55 * 0.25x

No. of women in the age group = 0.45 * 0.25x

Men not making a claim is 98% ,while women not making a claim is 98%

No. of men making claims = 0.98 * 0.55 * 0.25x

No. of women making claims = 0.98 * 0.45 * 0.25x

Total number of people making a policy-holders claim =0.98 * 0.55 * 0.25x + 0.98 * 0.45 * 0.25x .....(6)

We have to find out the probability of people not making claims in 60 nd above age group

Therefore , probabiity= no. of people not making claims in 60 and above age group / total no. of people not making claims in all age group.

From (4) ,(5) and (6)

probability = 0.90 * 0.55 * 0.25x + 0.95 * 0.45 * 0.25x /  0.90 * 0.55 * 0.25x + 0.95 * 0.45 * 0.25x + 0.96 * 0.50 * 0.50x + 0.97 * 0.50 * 0.50x + 0.98 * 0.55 * 0.25x + 0.98 * 0.45 * 0.25x

probability = 0.12375 x + 0.106875x / 0.12375 x + 0.106875x + 0.24x + 0.2425x + 0.13475x + 0.11025x

probability =0.230625 / 0.230625 + 0.4825 + 0.245

probability =0.230625 / 0.958125

probability = 0.2407

Therefore, probability of people not making claims are in 60 and above age group is 0.2407