SOA Exam P Question 119 | Conditional Probability of Joint Distribution

An auto insurance policy will pay for damage to both the policyholder’s car and
the other driver’s car in the event that the policyholder is responsible for an accident.
The size of the payment for damage to the policyholder’s car, X, has a marginal density
function of 1 for x E(0, 1]. Given X = x, the size of the payment for damage to the other
driver’s car, Y, has conditional density of 1 for y E[x, x + 1].
Given that the policyholder is responsible for an accident, calculate the probability that
the payment for damage to the other driver’s car will be greater than 0.5.
(A) 3/8
(B) 1/2
(C) 3/4
(D) 7/8
(E) 15/16

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[shirkycalypso]

When computing the prob(Y>.5) you can take the compliment P(Y>.5)= 1-P(Y<.5) and compute integral from 0 to .5 of y dy (since .5 is only defined on the part of Y from (0, 1)).
This gives you (y^2)/2 |(.5, 0) = 1/8-0= 1/8.
Plugging back in:
1 - 1/8 = 7/8

[miar]

this is so helpful. the sample solution didn't explain anything. thanks!

[miyukimatsune]

Can you explain why the second bound was to 2? Thanks!