8. An insurance company estimated that 30% of all automobile accidents were part

Question

8. An insurance company estimated that 30% of all automobile accidents were partly caused by weather conditions and that 20% of all automobile accidents involved bodily injury. Further, of those accidents that involved bodily injury, 40% were partly caused by weather conditions.

a. What is the probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury?

b. Are the events partly caused by weather conditions and involved bodily injury independent?

c. If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury?

d. What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury?

Explanation / Answer

Let W shows the event that automobile accident were partly caused by weather conditions and B shows the event that automobile accidents involved bodily injury. So from the given information we have

P(W) = 0.30, P(B) = 0.20

P(W|B) = 0.40

a)

The probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury is

P(W and B) = P(W|B)P(B) = 0.40*0.20 = 0.08

b)

No beucase P(W|B) = not equl to P(W)

c)

Here we need to find the probability P(B|W) so

P(B|W) = P(W and B) / P(W) = 0.08 / 0.30 = 0.2667

d)

By the addition rule we have

P(W or B) = P(W) + P(B) - P(W and B) = 0.3 +0.2 -0.08 = 0.42

So the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury is

P(not W and not B) = 1 - P(W or B) = 1 - 0.42 = 0.58

Related Questions

Navigate

• Browse (All)
• Browse 8
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.