Question

Calculate each binomial probability:

(a) Fewer than 5 successes in 11 trials with a 10 percent chance of success. (Round your answer to 4 decimal places.)

Probability =      

(b) At least 2 successes in 8 trials with a 30 percent chance of success. (Round your answer to 4 decimal places.)

Probability =   

(c) At most 10 successes in 18 trials with a 70 percent chance of success. (Round your answer to 4 decimal places.)

Probability =             

Answer 1

a)
Here, n = 11, p = 0.1, (1 - p) = 0.9 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) = (11C0 * 0.1^0 * 0.9^11) + (11C1 * 0.1^1 * 0.9^10) + (11C2 * 0.1^2 * 0.9^9) + (11C3 * 0.1^3 * 0.9^8) + (11C4 * 0.1^4 * 0.9^7)
P(X < 5) = 0.3138 + 0.3835 + 0.2131 + 0.071 + 0.0158
P(X < 5) = 0.9972

b)

Here, n = 8, p = 0.3, (1 - p) = 0.7 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 2).
P(X >= 2) = (8C2 * 0.3^2 * 0.7^6) + (8C3 * 0.3^3 * 0.7^5) + (8C4 * 0.3^4 * 0.7^4) + (8C5 * 0.3^5 * 0.7^3) + (8C6 * 0.3^6 * 0.7^2) + (8C7 * 0.3^7 * 0.7^1) + (8C8 * 0.3^8 * 0.7^0)
P(X >= 2) = 0.2965 + 0.2541 + 0.1361 + 0.0467 + 0.01 + 0.0012 + 0.0001
P(X >= 2) = 0.7447

c)

Here, n = 18, p = 0.7, (1 - p) = 0.3 and x = 10
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <= 10).
P(X <= 10) = (18C0 * 0.7^0 * 0.3^18) + (18C1 * 0.7^1 * 0.3^17) + (18C2 * 0.7^2 * 0.3^16) + (18C3 * 0.7^3 * 0.3^15) + (18C4 * 0.7^4 * 0.3^14) + (18C5 * 0.7^5 * 0.3^13) + (18C6 * 0.7^6 * 0.3^12) + (18C7 * 0.7^7 * 0.3^11) + (18C8 * 0.7^8 * 0.3^10) + (18C9 * 0.7^9 * 0.3^9) + (18C10 * 0.7^10 * 0.3^8)
P(X <= 10) = 0 + 0 + 0 + 0 + 0 + 0.0002 + 0.0012 + 0.0046 + 0.0149 + 0.0386 + 0.0811
P(X <= 10) = 0.1406