A fair die is rolled repeatedly. Let [math]X[/math] be the number of rolls needed to obtain a 5 and [math]Y[/math] the number of rolls needed to obtain a 6.

Calculate [math]\operatorname{E}(X | Y = 2). [/math]

• 5.0
• 5.2
• 6.0
• 6.6
• 6.8

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

[math]X[/math] follows a geometric distribution with [math]p = \frac{1}{6}[/math]. [math]Y = 2[/math] implies the first roll is not a 6 and the second roll is a 6. This means a 5 is obtained or the first time on the first roll (probability = 20%) or a 5 is obtained for the first time on the third or later roll (probability = 80%).

[[math]] \operatorname{E}[X | X \geq 3] = \frac{1}{p} + 2 = 6 + 2 = 8, [[/math]]

so [math]\operatorname{E}[X | Y = 2] [/math] equals

0.2(1) + 0.8(8) = 6.6. 

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.