Answer: ez!
Explanation:
To compute the probabilities, we can use the binomial probability formula:
P(x = k) = nCk * p^k * (1-p)^(n-k)
where n is the number of trials, p is the probability of success, k is the number of successes.
(a) P(x ≤ 2), n = 4, p = 0.2:
P(x ≤ 2) = P(x = 0) + P(x = 1) + P(x = 2)
P(x = 0) = 4C0 * 0.2^0 * 0.8^4 = 0.8^4 = 0.4096
P(x = 1) = 4C1 * 0.2^1 * 0.8^3 = 4 * 0.2 * 0.8^3 = 0.4096
P(x = 2) = 4C2 * 0.2^2 * 0.8^2 = 6 * 0.2^2 * 0.8^2 = 0.1536
P(x ≤ 2) = 0.4096 + 0.4096 + 0.1536 = 0.9728
(b) P(x > 4), n = 6, p = 0.8:
P(x > 4) = P(x = 5) + P(x = 6)
P(x = 5) = 6C5 * 0.8^5 * 0.2^1 = 6 * 0.8^5 * 0.2 = 0.3932
P(x = 6) = 6C6 * 0.8^6 * 0.2^0 = 0.8^6 = 0.2621
P(x > 4) = 0.3932 + 0.2621 = 0.6553
(c) P(x < 2), n = 9, p = 0.2:
P(x < 2) = P(x = 0) + P(x = 1)
P(x = 0) = 9C0 * 0.2^0 * 0.8^9 = 0.8^9 = 0.1342
P(x = 1) = 9C1 * 0.2^1 * 0.8^8 = 9 * 0.2 * 0.8^8 = 0.3874
P(x < 2) = 0.1342 + 0.3874 = 0.5216
(d) P(x ≥ 1), n = 6, p = 0.9:
P(x ≥ 1) = 1 - P(x = 0)
P(x = 0) = 6C0 * 0.9^0 * 0.1^6 = 0.9^0 * 0.1^6 = 0.000001
P(x ≥ 1) = 1 - 0.000001 = 0.999999