Question

An marine insurance company monitors the largest claim paid each month. Claims may be paid on three types of boats: fishing vessel, yacht and speed boat. The probability that the largest claim is from a yatch is 0.26, and that it is from a speed boat is 0.12, and that it is from a fishing vessel is 0.62. Calculate the probability that over a four month period, the largest claim is from a fishing vessel on a least 2 more occasions that from a yatch.

Answer 1

For the largest claim to be from a fishing vessel at least two more occasions from the yatch, we know that the difference can be 2,3,or 4.

Thus, we will have to find the probabilities of all three events individually and then add them in the end.

We will denote a yatch with Y, a speed boat with SB and a fishing vessel with FV.

P(difference is 4)

For the difference to be 4, all 4 claims must come from a fishing vessel. Thus, the required probability is

= 0.62*0.62*0.62*0.62

= 0.1477634

P(difference is 3)

For the difference to be 3, we know the following combinations can occur:

FV FV FV SB

FV FV SB FV

FV SB FV FV

SB FV FV FV

Thus, the required probability is

= (0.62*0.62*0.62*0.12)⁴

= 0.0000006689987

P(difference is 2)

This can occur either when there are 3 fishing vessels and 1 yatch or when there are 2 fishing vessels and 0 yatchs.

3 FV, 1 Y

FV FV FV SB

FV FV SB FV

FV SB FV FV

SB FV FV FV

Thus, the required probability is

= (0.62*0.62*0.62*0.26)⁴

= 0.00001474326

2 FV 2 SB

FV FV SB SB

FV SB FV SB

FV SB SB FV

SB FV FV SB

SB FV SB FV

SB SB FV FV

= (0.62*0.62*0.12*0.12)^6

= 0.000000000000002876572

Thus, the required probability is

= 0.000000000000002876572 + 0.00001474326 + 0.0000006689987 + 0.1477634

= 0.1477788 or 14.77%