A car insurance company has high-risk, medium-risk, and low-risk clients, who have, respectively, probabilities. 04,.02, and. 01 of filing claims within a given year. The proportions of the numbers of clients in the three categories are. 15,.25, and. 60, respectively. (a) What is the probability that a random client doesn't file a claim? (b) What proportion of the claims filed each year come from high-risk clients? (c) What is the probability that a random client who didn't file a claim is low - risk?

(a) 0.983

(b) 0.353 or 35.3%

(c) 0.604 or 60.4%

Step-by-step explanation:

a) The probability of a random client does not file a claim is equal to the sum of:

1) the probability of a client being high risk and does not file a claim = P (hr) * (1-P (c_hr))

2) the probability of a client being medium risk and does not file a claim = P (mr) * (1-P (c_mr))

and

3) the probability of a client being low risk and does not file a claim = P (lr) * (1-P (c_lr))

P (not claim) = P (hr) * (1-P (c_hr)) + P (mr) * (1-P (c_mr)) + P (lr) * (1-P (c_lr))

P (not claim) = 0.15 * (1-0.04) + 0.25 * (1-0.02) + 0.6 * (1-0.01)

P (not claim) = 0.15*0.96+0.25*0.98+0.6*0.99 = 0.983

(b) To know the proportion of claims that come from high risk clients we need to know the total expected claims in every category:

Claims expected by high risk clients = P (c_hr) * P (hr) = 0.04*0.15 = 0.006 claims/client

Claims expected by medium risk clients = P (c_mr) * P (mr) = 0.02*0.25 = 0.005 claims/client

Claims expected by low risk clients = P (c_lr) * P (lr) = 0.01*0.60 = 0.006 claims/client

The proportion of claims done by high risk clients is

Claims by HR clients / Total claims expected = 0.006 / (0.006+0.005+0.006) = 0.006 / 0.017 = 0.3529 or 35,3%

(c) The probability of being a client of a particular category and who don't file a claim is:

1) High risk: 0.15 * (1-0.04) = 0.144

2) Medium risk: 0.25 * (1-0.02) = 0.245

3) Low risk: 0.6 * (1-0.01) = 0.594

The probability that a random client who didn't file a claim is low - risk can be calculated as:

Probability of being low risk and don't file a claim / Probability of not filing a claim

P (LR¬ claim) / P (not claim) = 0.594 / (0.144+0.245+0.594)

P (LR¬ claim) / P (not claim) = 0.594 / 0.983 = 0.604 or 60.4%