The Strachovania Insurance Company offers quotations for mot

The Strachovania Insurance Company offers quotations for motor insurance by telephone. The company employs permanent staff to do this work but as a result of dramatic rise in the number of telephone enquiries 35% of quotations are provided by temporary staff. Unfortunately 22% of the quotations provided by temporary staff prove to be wrong, compared to the 8% of the quotations provided by full-time staff that turn out to be wrong. Under the contract with the agency supplying the temporary staff, the agency will pay a proportion of the total costs of mistakes based on the proportion of them that are made by the temporary staff. Use Bayes\' rule to determine the probability that if a mistake has been made by one of the temporary staff and use it to suggest what proportion of the total costs of mistakes the agency should pay.

Solution

temporary staff    0.35

full time staff    0.65

wrong of temprary staff    0.22

wrong of full time staff   0.08

probability that if a mistake has been made by one of the temporary staff

P( temporary staff / mistake)

P( mistake) = 0.08*0.65   + 0.22*0.35   = 0.129

P( temporary staff / mistake) = 0.22*0.35 / 0.129 = 0.5969

what proportion of the total costs of mistakes the agency should pay.

0.22 * 0.35 = 0.077