Question

Suppose that 20% of all college students smoke cigarettes. A sample of 12 is selected randomly. What is the probability that less than 8 students smoke? Round your answer to four decimal places.

Answer 1

Solution

Given that ,

p = 0.20

1 - p = 0.80

n = 12

Using binomial probability formula ,

P(X = x) = ((n! / x! (n - x)!) * p^x * (1 - p)^{n - x}

P(X < 8) = P(X = 0) +P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

P(X < 8 ) = ((12! / 0! (12 - 0)!) * 0.20⁰ * (0.80)^{12 - 0} +  ((12! / 1! (12 - 1)!) * 0.20¹ * (0.80)^{12 - 1}

((12! / 2! (12 - 2)!) * 0.20² * (0.80)^{12 - 2}+  ((12! / 3! (12 - 3)!) * 0.20³ * (0.80)^{12 - 3}+ ((12! / 4! (12 - 4)!) * 0.20⁴ * (0.80)^{12 - 4}+ ((12! / 5! (12 - 5)!) * 0.20⁵ * (0.80)^{12 - 5}+ ((12! /6! (12 - 6)!) * 0.20⁶ * (0.80)^{12 - 6}+ ((12! / 7! (12 - 7)!) * 0.20⁷ * (0.80)^{12 - 7}

=  0.0687 + 0.2062 + 0.2835 + 0.2362 + 0.1329 + 0.0532 + 0.0155 + 0.0033

Probability = 0.9994