*Let A = Low and B = 3 vehicles involved. ?? (??|??) = ?? (??|??) = ?? (?? ?????

Question

*Let A = Low and B = 3 vehicles involved. ?? (??|??) = ?? (??|??) = ?? (?? ?????? ??) = Are the events A, B independent? Use the conditional multiplication formula and the product rule.

*Let A = Medium and B = 2 vehicles involved ?? (??|??) = ?? (??|??) = ?? (?? ?????? ??) = Are the events A, B independent? Use the conditional multiplication formula and the product rule.

Use this information to answer questions 3-4. The data for Claim Severity and Vehicles involved are summarized on Vehicles involved more the right. Add totals first. Low Medium High 30 15 24 12 4 rD

Explanation / Answer

Case 1: A = low and B =3 vehicles involved

P(A) = 60/100 = 0.6

P(B) = 40/100 = 0.4

P(A and B) = 24/100 = 0.24

Using product rule, if P(A and B) = P(A).P(B), then A and B are independent.

P(A).P(B) = 0.6×0.4 = 0.24 = P(A and B)

And from conditional multiplication rule,

P(B|A) = P( A and B)/ P(A) = 0.24/0.6 = 0.4 = P(A)

Thus, A and B are independent

Case 2: A =Medium, B =2 vehicles involved

P(A) = 28/100 = 0.28

P(B) = 50/100 = 0.5

P( A and B) = 15/100 = 0.15

Using product rule,

P(A).P(B) = 0.28×0.5 = 0.14 not equal to P(A and B)

Using conditional multiplication rule,

P(B|A) = P(A and B)/P(A) = 0.15/0.28 = 15/28 not equal to P(B)

Thus, A and B are not independent in this case

2 3 more Total Low 30 24 6 60 Medium 15 12 1 28 High 5 4 3 12 Total 50 40 10 100

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