Question

Fifty-nine percent of US adults have little confidence in their cars. You randomly select eight US adults. Find the probability that the number of US adults who have little confidence in their cars is (1) exactly three and then find the probability that it is (2) more than 6. show the work please

Answer 1

Let X be the number out of 8 US adults who have little confidence in their cars. Then X ~ Binomial( p = 0.59, n = 8)

(1)

Probability that the number of US adults who have little confidence in their cars is exactly three

= P(X = 3) = ⁸C₃ * 0.59³ * (1 - 0.59)^8-3

= 8! / (8-3)! 3! * 0.59³ * 0.41⁵

= 56 * 0.59³ * 0.41⁵

= 0.1332

(2)

Probability that the number of US adults who have little confidence in their cars is more than six

= P(X > 6) = P(X = 7) + P(X = 8) = ⁸C₇ * 0.59⁷ * (1 - 0.59)^8-7  + ⁸C₈ * 0.59⁸ * (1 - 0.59)^8-8

= 8! / (8-7)! 7! * 0.59⁷ * 0.41 + 8! / (8-8)! 8! * 0.59⁸

= 8 * 0.59⁷ * 0.41 + 0.59⁸

= 0.0963