Consider a binomial experiment with 5 trials and p=0.4.a. Compute the probability of 1 success, f(1).b. Compute f(0).c. Compute f(2).d. Find the probability of at least one success.e. Find the expected value, variance, and standard deviation

a. To compute the probability of 1 success, f(1), we can use the formula for the binomial probability:

f(1) = (5 choose 1) * (0.4)^1 * (0.6)^(5-1)

Using the formula for combinations, (5 choose 1) = 5, we can substitute the values into the formula:

f(1) = 5 * (0.4)^1 * (0.6)^4

Calculating this expression, we find that f(1) = 0.3456.

b. To compute f(0), the probability of 0 successes, we can use the same formula:

f(0) = (5 choose 0) * (0.4)^0 * (0.6)^(5-0)

Since (5 choose 0) = 1, we can simplify the formula:

f(0) = 1 * (0.4)^0 * (0.6)^5

Calculating this expression, we find that f(0) = 0.07776.

c. To compute f(2), the probability of 2 successes, we can again use the binomial probability formula:

f(2) = (5 choose 2) * (0.4)^2 * (0.6)^(5-2)

Using the formula for combinations, (5 choose 2) = 10, we can substitute the values into the formula:

f(2) = 10 * (0.4)^2 * (0.6)^3

Calculating this expression, we find that f(2) = 0.3456.

d. To find the probability of at least one success, we can calculate the complement of the probability of 0 successes:

P(at least one success) = 1 - f(0)

Substituting the value of f(0) we calculated earlier, we have:

P(at least one success) = 1 - 0.07776

Calculating this expression, we find that P(at least one success) = 0.92224.

e. To find the expected value, variance, and standard deviation, we can use the formulas for a binomial distribution:

Expected value (mean) = n * p

Variance = n * p * (1 - p)

Standard deviation = sqrt(Variance)

Substituting the values n = 5 and p = 0.4 into the formulas, we have:

Expected value = 5 * 0.4 = 2

Variance = 5 * 0.4 * (1 - 0.4) = 1.2

Standard deviation = sqrt(1.2) ≈ 1.095

Therefore, the expected value is 2, the variance is 1.2, and the standard deviation is approximately 1.095.