(# 42 from the SOA list, with two slight changes.) For Company A there is a 70%

Question

(# 42 from the SOA list, with two slight changes.) For Company A there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 10,000 and standard deviation 2,000. For Company B there is a 60% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 9,000 and standard deviation 2,000. The total claim amounts of the two c Calculate the probability that, in the coming year, Company B's total claim amount will exceed Company A's total claim amount. (A) 0.280 (B) 0.285 (C) 0.317 (D) 0.323 (E) 0.340

Explanation / Answer

P(B > A)

there are two case when it is possible

1) when B makes a claim and A does not

2) and when both A and B makes claim

probability of event 1 = (1-0.6) * 0.7 = 0.28

probability of event 2 = .0.4*0.3 = 0.12

but P(B > A|event 2)

B -A follow N(9000 - 10000, 2000^2 * 2)

P(B -A >0)

= P(Z > (+1000/(2000*sqrt(2)))

=P(Z> 0.3535) )

= 0.3619

hence

total probability

= 0.28 +0.3619*0.12

= 0.323428

option D) is correct

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