Final answer:
To compute P(x), use the binomial probability formula. Compute P(x≤2), P(x>2), P(x<6), and P(x≥1) for given n and p values using the binomial probability formula. Therefore we get is 0.652.
Step-by-step explanation:
To compute P(x), we can use the binomial probability formula. The formula is:
P(x) = C(n, x) * p^x * (1-p)^(n-x)
Let's compute P(x) for each case:
(a) P(x≤2), n=6, p=0.8
P(x≤2) = P(x=0) + P(x=1) + P(x=2)
P(x=0) = C(6, 0) * (0.8)^0 * (1-0.8)^(6-0) = 0.001
P(x=1) = C(6, 1) * (0.8)^1 * (1-0.8)^(6-1) = 0.010
P(x=2) = C(6, 2) * (0.8)^2 * (1-0.8)^(6-2) = 0.046
P(x≤2) = 0.001 + 0.010 + 0.046 = 0.057
So, P(x≤2) = 0.057
(b) P(x>2), n=3, p=0.6
P(x>2) = 1 - P(x≤2)
P(x≤2) = P(x=0) + P(x=1) + P(x=2)
P(x=0) = C(3, 0) * (0.6)^0 * (1-0.6)^(3-0) = 0.064
P(x=1) = C(3, 1) * (0.6)^1 * (1-0.6)^(3-1) = 0.288
P(x=2) = C(3, 2) * (0.6)^2 * (1-0.6)^(3-2) = 0.432
P(x≤2) = 0.064 + 0.288 + 0.432 = 0.784
So, P(x>2) = 1 - 0.784 = 0.216
(c) P(x<6), n=7, p=0.1
P(x<6) = P(x≤5)
P(x≤5) = P(x=0) + P(x=1) + P(x=2) + P(x=3) + P(x=4) + P(x=5)
P(x=0) = C(7, 0) * (0.1)^0 * (1-0.1)^(7-0) = 0.478
P(x=1) = C(7, 1) * (0.1)^1 * (1-0.1)^(7-1) = 0.354
P(x=2) = C(7, 2) * (0.1)^2 * (1-0.1)^(7-2) = 0.207
P(x=3) = C(7, 3) * (0.1)^3 * (1-0.1)^(7-3) = 0.072
P(x=4) = C(7, 4) * (0.1)^4 * (1-0.1)^(7-4) = 0.015
P(x=5) = C(7, 5) * (0.1)^5 * (1-0.1)^(7-5) = 0.002
P(x≤5) = 0.478 + 0.354 + 0.207 + 0.072 + 0.015 + 0.002 = 1.128
So, P(x<6) = P(x≤5) = 1.128
(d) P(x≥1), n=9, p=0.1
P(x≥1) = 1 - P(x=0)
P(x=0) = C(9, 0) * (0.1)^0 * (1-0.1)^(9-0) = 0.348
P(x≥1) ≈ 1 - 0.348 = 0.652