Final answer:
To solve for various probabilities and statistics in a binomial distribution with n = 13 and p = 0.3, we use the binomial probability formula. We can calculate f(0), f(12), p(x≤1), p(x≥3), E(x), Var(x), and sigma. The calculated probabilities and expected value, variance, and standard deviation provide key statistics about the distribution.
Step-by-step explanation:
To solve the binomial probability questions, we will use the binomial probability formula,
f(x) = (n choose x) · p^x · (1-p)^(n-x),
where n is the number of trials, p is the probability of success on a single trial, and x is the number of successes we want to calculate the probability for.
a. To compute f(0), the probability of 0 successes in 13 trials (with p = 0.3), we plug the values into the formula:
f(0) = (13 choose 0) · 0.3^0 · (1-0.3)^(13-0) = 1 · 1 · (0.7)^13 ≈ 0.0067
b. To compute f(12), the probability of 12 successes in 13 trials, we use the formula:
f(12) = (13 choose 12) · 0.3^12 · (1-0.3)^(13-12) ≈ 0.0000
c. To compute p(x≤1), we add the probabilities of getting 0 or 1 successes:
p(x≤1) = f(0) + f(1), where f(1) = (13 choose 1) · 0.3^1 · (1-0.3)^(13-1) , so we compute f(1) and sum it with f(0).
d. To compute p(x≥3), we use either the complement rule or sum the probabilities from 3 to 13, which can be tedious. The complement rule is p(x≥3) = 1 - p(x≤2), and we calculate p(x≤2) similarly to part (c).
e. To compute the expected value E(x), we use the formula E(x) = n · p = 13 · 0.3 = 3.9, which is the mean number of successes.
f. To compute variance Var(x) and standard deviation sigma, we use the formulas:
Var(x) = n · p · q = 13 · 0.3 · (1-0.3) ≈ 2.73
sigma = √Var(x) ≈ 1.65, where q is the probability of failure (1-p).