An urn contains four fair dice. Two have faces numbered 1, 2, 3, 4, 5, and 6; one has faces numbered 2, 2, 4, 4, 6, and 6; and one has all six faces numbered 6. One of the dice is randomly selected from the urn and rolled. The same die is rolled a second time. Calculate the probability that a 6 is rolled both times.

• 0.174
• 0.250
• 0.292
• 0.380
• 0.417

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

Solution: C

Let event A be the selection of the die with faces (1,2,3,4,5,6), event B be the selection of the die with faces (2,2,4,4,6,6) and event C be the selection of the die with all 6’s. The desired probability is, using the law of total probability,

[[math]] \begin{align*} \operatorname{P}(6, 6) &= \operatorname{P}(6, 6 | A) \operatorname{P}( A) + \operatorname{P}(6, 6 | B) \operatorname{P}( B) + \operatorname{P}(6, 6 | C ) \operatorname{P}(C ) \\ &= (1/ 36)(1/ 2) + (1/ 9)(1/ 4) + 1(1/ 4) \\ &=1/ 72 + 2 / 72 + 18 / 72 = 21/ 72 \\ &= 0.292. \end{align*} [[/math]]

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