In a study of fatal car accidents, some accidents were attributed to high speed or drunk driving. a. the probability that the fatality was attributed to high speeds was 0.57. b. the probability that the fatality was attributed to drunk driving was 0.62. c. the probability that high speed and drunk driving caused the accident was 0.41. a. [2 pts] what is the probability that a fatal accident will be attributed to high speeds or drunk driving? b. [2 pts] what is the probability that a fatal accident will be attributed to neither cause? c. [2 pts] what is the probability that the fatality will be attributed to drunk driving, but not high speeds?

a) Probability that a fatal accident will be attributed to high speeds or drunk driving is 0.37

b) Probability that a fatal accident will be attributed to neither cause is 0.22

c) Probability that the fatality will be attributed to drunk driving, but not high speeds is 0.21

Explanation:

The easiest way to solve this problem is to make a venn diagram to see the distribution of probability as it is in the image. For this you first need to establish the variables.

P(h) = Probability that a fatal accident will be attributed to high speeds

P(d) = Probability that a fatal accident will be attributed to drunk driving

P(hnd) = Probability that high speed and drunk driving caused the accident

P(hud) = Probability that a fatal accident will be attributed to high speeds or drunk driving

So we have the values for the following variables

P(h) = 0.57     P(d) = 0.62   P(hnd) = 0.41

1. In the diagram you can see that P(h) and P(d) include the probability of both so to calculate the probability  that a fatal accident will be attributed to high speeds or drunk driving we need to take out the intersection for each and add the result then:

P(hud) = (P(h) -  P(hnd)) + (P(d) - P(hnd)

P(hud) = (0.57 - 0.41) + (0.62 - 0.41)

P(hud) = 0.16 + 0.21

P(hud) = 0.37

So the probability that a fatal accident will be attributed to high speeds or drunk driving is 0.37

2. To calculate the probability that a fatal accident will be attributed to neither cause we need to substrat from the total probability that is 1 the probability of P(hud) with the probability of P(hnd) so we have all the information that is en each part of the diagram.

Probability of neither cause = 1.0 - (P(hud) + P(hnd))

Probability of neither cause = 1.0 - (0.37 + 0.41)

Probability of neither cause = 1.0 - 0.78

Probability of neither cause = 0.22

In the diagram it is what is out of the circles.

So the probability that a fatal accident will be attributed to neither cause is 0.22

3. To calculate the probability that the fatality will be attributed to drunk driving, but not high speeds we have to subtract to P(d), the probability of the intersection, P(hnd)

Probability just to drunk driving = P(d) - P(hnd)

Probability just to drunk driving = 0.62 - 0.41

Probability just to drunk driving = 0.21

In the diagram is what appears in the right of the red circle.

So the probability that the fatality will be attributed to drunk driving, but not high speeds is 0.21