Question

Find the mean and standard deviation of the number of 4’s obtained in 12 tosses of a fair die.

Answer 1

To find the mean and standard deviation of the number of 4's obtained in 12 tosses of a fair die, we calculate the probability distribution using a binomial distribution with n = 12 and p = 1/6. The mean is 2 and the standard deviation is approximately 1.83.

Step-by-step explanation:

To find the mean and standard deviation of the number of 4's obtained in 12 tosses of a fair die, we need to consider the probability distribution of getting a 4 on a single toss.

The probability of getting a 4 on a fair die is 1/6. Since each toss is independent, the probability distribution will follow a binomial distribution with n = 12 and p = 1/6.

The mean of a binomial distribution is given by np, so the mean number of 4's out of 12 tosses is (12)(1/6) = 2.

The standard deviation of a binomial distribution is given by sqrt(npq), where q = 1 - p. So the standard deviation of the number of 4's out of 12 tosses is sqrt((12)(1/6)(5/6)) = sqrt(10/3) ≈ 1.83.

Answer 2

The mean number of 4's obtained in 12 tosses of a fair die is 2, and the standard deviation is the square root of 2.

Step-by-step explanation:

To find the mean and standard deviation of the number of 4's obtained in 12 tosses of a fair die, we use the properties of a binomial distribution. When rolling a die, getting a 4 can be considered a 'success.' Since we have a six-sided die, the probability, p, of getting a 4 on any single toss is 1/6, and the probability of not getting a 4, q, is 5/6.

The mean (μ) of a binomial distribution is calculated as μ = n * p, where n is the number of trials, which in this case is 12. Plugging in the numbers:

μ = 12 * (1/6) = 2

The formula for the standard deviation (σ) of a binomial distribution is σ = sqrt(n * p * q). Plugging in our values:

σ = sqrt(12 * (1/6) * (5/6))
= sqrt(2)

Therefore, the mean number of 4's is 2 and the standard deviation is the square root of 2.