A dice is rolled three times and the sum of three numbers appearing on the uppermost face is 15. What is the chance that the first roll was four?

To solve this problem, we need to understand the possible outcomes when a dice is rolled three times and the sum of the numbers is 15.

Step 1: Identify the possible outcomes The possible outcomes when a dice is rolled three times and the sum is 15 are (6,6,3), (6,5,4), (5,5,5), (6,4,5), (5,6,4), (4,6,5), and their permutations.

Step 2: Identify the outcomes where the first roll was four The outcomes where the first roll was four are (4,6,5), (4,5,6), and their permutations.

Step 3: Calculate the probability The probability that the first roll was four is the number of outcomes where the first roll was four divided by the total number of outcomes.

There are 2 permutations each for (4,6,5) and (4,5,6), so there are 4 outcomes where the first roll was four.

There are 3 permutations each for (6,6,3), (6,5,4), (5,5,5), (6,4,5), (5,6,4), and (4,6,5), so there are 18 total outcomes.

Therefore, the probability that the first roll was four is 4/18 = 2/9.