Question

70% of the students applying to a university are accepted. Assume the requirements for a binomial experiment are satisfied for 10 applicants.

a. What is the probability that among the next 10 applicants 8 or more will be accepted.

b. What is the probability that among the next 10 applicants 4 or more will be accepted? (Use the binomial table for this problem)

c. What is the expected number of the next 10 applicants that will be accepted?

Answer 1

a)
Here, n = 10, p = 0.7, (1 - p) = 0.3 and x = 8
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 8).
P(X >= 8) = (10C8 * 0.7^8 * 0.3^2) + (10C9 * 0.7^9 * 0.3^1) + (10C10 * 0.7^10 * 0.3^0)
P(X >= 8) = 0.2335 + 0.1211 + 0.0282
P(X >= 8) = 0.3828

b)
We need to calculate P(X >= 4).
P(X >= 4) = (10C4 * 0.7^4 * 0.3^6) + (10C5 * 0.7^5 * 0.3^5) + (10C6 * 0.7^6 * 0.3^4) + (10C7 * 0.7^7 * 0.3^3) + (10C8 * 0.7^8 * 0.3^2) + (10C9 * 0.7^9 * 0.3^1) + (10C10 * 0.7^10 * 0.3^0)
P(X >= 4) = 0.0368 + 0.1029 + 0.2001 + 0.2668 + 0.2335 + 0.1211 + 0.0282
P(X >= 4) = 0.9894

c)
expected number = 10 * 0.7 = 7