A box contains 6 black, 5 brown and 2 yellow balls. If 2 balls are selected at random, what is the probability that both are black?

To solve this problem, we need to determine the probability of selecting 2 black balls out of the total number of balls in the box.

Step 1: Calculate the total number of balls. Total balls = 6 black + 5 brown + 2 yellow = 13 balls.

Step 2: Calculate the number of ways to choose 2 balls out of 13. The number of ways to choose 2 balls out of 13 is given by the combination formula: C(n, k) = n! / [k!(n - k)!] where n is the total number of items, and k is the number of items to choose. C(13, 2) = 13! / [2!(13 - 2)!] = 13! / (2! * 11!) = (13 * 12) / (2 * 1) = 78

Step 3: Calculate the number of ways to choose 2 black balls out of 6. C(6, 2) = 6! / [2!(6 - 2)!] = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15

Step 4: Calculate the probability of both balls being black. The probability is the number of favorable outcomes divided by the total number of possible outcomes. Probability = Number of ways to choose 2 black balls / Number of ways to choose 2 balls out of 13 Probability = 15 / 78

Step 5: Simplify the fraction. 15 / 78 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. 15 / 78 = (15 ÷ 3) / (78 ÷ 3) = 5 / 26

Therefore, the probability that both balls selected are black is 5/26.