Answer:
(a) 0.983
(b) 0.353 or 35.3%
(c) 0.604 or 60.4%
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%