Question

Airbags are manufactured by Aces (A), Best (B), and Cool (C) at rates of 57%, 26% and 17%, respectively. Airbags occasionally kill (K) passengers when they deploy in accidents. Airbags made by Aces, Best, and Cool do not kill people at rates of 99%, 96%, and 87%, respectively. One airbag is randomly selected for testing.

1) Express the airbag manufacturing data as probabilities. Find the probability that a randomly selected airbag was manufactured by Best.

2) Express the airbag fatality data as probabilities. Find the probability that a randomly selected airbag killed the passenger given that it was manufactured by Cool.

3) If an airbag kills a passenger, calculate the probability that the airbag was manufactured by Cool.

Answer 1

P(Aces) = 0.57

P(Best) = 0.26

P(Cool) = 0.17

P(do not kill | ace) = 0.99

P(do not kill | best) = 0.96

P(do not kill | cool) = 0.87

1) P(manufactured by best) = P(best) = 0.26

2) P(killed | cool) = 1 - P(do not kill | cool) = 1 - 0.87 = 0.13

P(killed | ace) = 1 - P(do not kill | ace) = 1 - 0.99 = 0.01

P(killed | best) = 1 - P(do not kill | best) = 1 - 0.96 = 0.04

P(killed) = P(killed | ace) * P(ace) + P(killed | best) * P(best) + P(killed | cool) * P(cool)

              = 0.01 * 0.57 + 0.04 * 0.26 + 0.13 * 0.17

              = 0.0382

P(cool | killed) = P(killed | cool) * P(cool) / P(killed)

                        = 0.13 * 0.17 / 0.0382

                        = 0.5785