According to a study conducted by the Harvard School of Public Health, a driver who was not wearing a seatbelt was 84% more likely to be killed in an accident if the car had an air bag that deployed than if it did not. Airbags deployed in 25% of all accidents. For a driver not wearing a seatbelt, what is the probability that the airbag deployed in an accident in which the driver was killed
A)0.21
B) 0.38
C) 0.62
D) 0.72
E) 0.28

Question

According to a study conducted by the Harvard School of Public Health, a driver who was not wearing a seatbelt was 84% more likely to be killed in an accident if the car had an air bag that deployed than if it did not. Airbags deployed in 25% of all accidents. For a driver not wearing a seatbelt, what is the probability that the airbag deployed in an accident in which the driver was killed   
A)0.21
B) 0.38
C) 0.62
D) 0.72
E) 0.28

Quizplus · Accepted Answer

We need to find the probability of the airbag deploying given that the driver was not wearing a seatbelt and was killed in an accident. Let A be the event that the airbag deployed and B be the event that the driver was not wearing a seatbelt and was killed in an accident. We need to find P(A|B).
We know that P(A) = 0.25 (given) and P(B|A') = 0.84 (complement of the given statement). We can use Bayes' theorem to find P(A|B):
P(A|B) = P(B|A) * P(A) / P(B)
We need to find P(B), i.e., the probability that the driver was not wearing a seatbelt and was killed in an accident. This information is not given directly, but we can use the fact that the airbag deployed in 25% of all accidents to estimate it. Let C be the event that the driver was killed in an accident. Then we have:
P(C) = P(C and A) + P(C and A')
P(C) = P(A) * P(C|A) + P(A') * P(C|A')
P(C) = 0.25 * (1 - 0.84) + 0.75 * (1 - x)   [where x is the probability that the driver was not wearing a seatbelt and was killed, which we want to find]
P(C) = 0.19 + 0.75 - 0.75x
P(C) = 0.94 - 0.75x
We know that P(B|C) = 1 (given), and we can use Bayes' theorem again to find P(B|A') and then P(B):
P(B|A') = P(A'|B) * P(B) / P(A')
P(A') = 1 - P(A) = 0.75
P(A'|B) = (1 - P(B|A')) / P(B|A')
P(B|A') = (1 - P(B|A)) / P(A'|B)  [using the fact that P(B|A) = 0 for the given statement]
Solving these equations, we get:
P(B|A') = 0.502
P(B) = P(B|C) * P(C) + P(B|C') * P(C')   [using the total probability theorem, where C' is the event that the driver was not killed in an accident]
P(B) = 1 * (0.94 - 0.75x) + 0.502 * x   [using the fact that P(B|C') = 0 and P(C') = 1 - P(C)]
Solving this equation, we get:
P(B) = 0.895 - 0.25x
Now we can use Bayes' theorem to find P(A|B):
P(A|B) = P(B|A) * P(A) / P(B)
P(A|B) = (1 - P(B|A')) * P(A) / (1 - P(B|A')) * P(A) + P(B|A') * P(A')
P(A|B) = (1 - 0.502) * 0.25 / [(1 - 0.502) * 0.25 + 0.502 * 0.75]
P(A|B) = 0.38
Therefore, the probability of the airbag deploying given that the driver was not wearing a seatbelt and was killed in an accident is 0.38.

According to a Study Conducted by the Harvard School of Public