Question

According to a recent survey, 20% of college students purchase their lunch in the cafeteria. Assume 5 students are randomly selected. Use the binomial table in the text to answer the following questions.

a. What is the probability that all 5 eat in the cafeteria?

b. What is the probability that at least 3 eats in the cafeteria?

c. What is the probability that not more than 4 eat in the cafeteria?

d. What is the probability that exactly 1 student eats in the cafeteria?

Answer 1

a)

Here, n = 5, p = 0.2, (1 - p) = 0.8 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) = 5C5 * 0.2^5 * 0.8^0
P(X = 5) = 0.0003

b)

Here, n = 5, p = 0.2, (1 - p) = 0.8 and x = 3
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 3).
P(X >= 3) = (5C3 * 0.2^3 * 0.8^2) + (5C4 * 0.2^4 * 0.8^1) + (5C5 * 0.2^5 * 0.8^0)
P(X >= 3) = 0.0512 + 0.0064 + 0.0003
P(X >= 3) = 0.0579

c)

Here, n = 5, p = 0.2, (1 - p) = 0.8 and x = 3
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <= 3).
P(X <= 3) = (5C0 * 0.2^0 * 0.8^5) + (5C1 * 0.2^1 * 0.8^4) + (5C2 * 0.2^2 * 0.8^3) + (5C3 * 0.2^3 * 0.8^2)
P(X <= 3) = 0.3277 + 0.4096 + 0.2048 + 0.0512
P(X <= 3) = 0.9933

d)

Here, n = 5, p = 0.2, (1 - p) = 0.8 and x = 1
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 1)
P(X = 1) = 5C1 * 0.2^1 * 0.8^4
P(X = 1) = 0.4096