Final answer:
To solve for n, the number of dice rolled such that the probability of exactly two of them showing a number other than 1 is 25/216, we can use the binomial probability formula. Testing various values of n, we find that n = 4 satisfies the given probability.
Step-by-step explanation:
To find n, the number of fair 6-sided dice rolled such that the probability of exactly two of them showing a number other than 1 is 25/216, we must use the concepts of combinatorics and probability. Since each die has a 1/6 chance of landing on a specific number, the chance of rolling a number other than 1 is 5/6, and the probability of rolling a 1 is 1/6.
Let's assume that we roll n dice. We want exactly two dice to show a number other than 1, and the rest to show 1. The probability of this happening is calculated by:
The total probability is given by the formula:
P(exactly two not 1) = C(n, 2) * (5/6)^2 * (1/6)^(n-2)
We are told that P(exactly two not 1) = 25/216. We then solve the equation:
25/216 = C(n, 2) * (5/6)^2 * (1/6)^(n-2)
By testing different values of n, we find that n = 4 satisfies the equation:
25/216 = (6/1) * (25/36) * (1/36) = 25/216