Final answer:
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.