Question

Let x be a binomial random variable with n=7 and p=0.7. Find the following.

P(X = 4)

P(X < 5)

P(X ≥ 4)

Answer 1

a)

Here, n = 7, p = 0.7, (1 - p) = 0.3 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 4)
P(X = 4) = 7C4 * 0.7^4 * 0.3^3
P(X = 4) = 0.2269

b)

Here, n = 7, p = 0.7, (1 - p) = 0.3 and x = 5
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X < 5).
P(X < 5) = (7C0 * 0.7^0 * 0.3^7) + (7C1 * 0.7^1 * 0.3^6) + (7C2 * 0.7^2 * 0.3^5) + (7C3 * 0.7^3 * 0.3^4) + (7C4 * 0.7^4 * 0.3^3)
P(X < 5) = 0.0002 + 0.0036 + 0.025 + 0.0972 + 0.2269
P(X < 5) = 0.3529

c)

Here, n = 7, p = 0.7, (1 - p) = 0.3 and x = 4
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 4).
P(X >= 4) = (7C4 * 0.7^4 * 0.3^3) + (7C5 * 0.7^5 * 0.3^2) + (7C6 * 0.7^6 * 0.3^1) + (7C7 * 0.7^7 * 0.3^0)
P(X >= 4) = 0.2269 + 0.3177 + 0.2471 + 0.0824
P(X >= 4) = 0.8741