Question

52% of all Americans are home owners. If 45 Americans are randomly selected, find the probability that a. Exactly 21 of them are are home owners. b. At most 24 of them are are home owners. c. At least 23 of them are home owners. d. Between 20 and 24 (including 20 and 24) of them are home owners.

Answer 1

0.0918 ; 0.6277 ; 0.6064; 0.5055

Explanation:

Given that :

Proportion of home owners : p = 0.52

Number of trials = 45

Using binomial distribution formula :

P(x =x) = nCr * p^x * (1 - p)^(n-x)

A.) probability that exactly 21 are homeowners :

P(x = 21) = 45C21 * 0.52^21 * (1 - 0.52)^(45 - 21)

P(x = 21) = 45C21 * 0.52^21 * 0.48^24

P(x = 21) = 3773655750150 * 0.52^21 * 0.48^24

P(x = 21) = 0.0918

b. At most 24 of them are are home

Using the binomial probability to save computation time.

P(x ≤ 24) = p(0) + p(1) +.... + p(24)

P(x ≤ 24) = 0.6277

C.) At least 23 of them are home owners.

P(x ≥ 23) = p(23) + p(24) +... + p(45)

P(x ≥ 23) = 0.6064

d. Between 20 and 24 (including 20 and 24) of them are home owners.

P(20 ≤ x ≤ 24) = p(20) + p(21) + p(22) + p(23) + p(24)

P(20 ≤ x ≤ 24) = 0.0711 + 0.0918 + 0.1084 + 0.1175 + 0.1167

P(20 ≤ x ≤ 24) = 0.5055