Question

A large pool of adults earning their first driver’s license includes 50% low-risk drivers, 30% moderate-risk drivers, and 20% high-risk drivers. Because these drivers have no prior driving record, an insurance company considers each driver to be randomly selected from the pool. This month, the insurance company writes four new policies for adults earning their first driver’s license. Calculate the probability that these four will contain at least two more high-risk drivers than low-risk drivers.

Answer 1

The probability that these four will contain at least two more high-risk drivers than low-risk drivers is 0.0488.

Explanation:

Denote the different kinds of drivers as follows:

L = low-risk drivers

M = moderate-risk drivers

H = high-risk drivers

The information provided is:

P (L) = 0.50

P (M) = 0.30

P (H) = 0.20

Now, it given that the insurance company writes four new policies for adults earning their first driver’s license.

The combination of 4 new drivers that satisfy the condition that there are at least two more high-risk drivers than low-risk drivers is:

S = {HHHH, HHHL, HHHM, HHMM}

Compute the probability of the combination {HHHH} as follows:

P (HHHH) = [P (H)]⁴

= [0.20]⁴

= 0.0016

Compute the probability of the combination {HHHL} as follows:

P (HHHL) =
`{4\choose 1}` × [P (H)]³ × P (L)

= 4 × (0.20)³ × 0.50

= 0.016

Compute the probability of the combination {HHHM} as follows:

P (HHHL) =
`{4\choose 1}` × [P (H)]³ × P (M)

= 4 × (0.20)³ × 0.30

= 0.0096

Compute the probability of the combination {HHMM} as follows:

P (HHMM) =
`{4\choose 2}` × [P (H)]² × [P (M)]²

= 6 × (0.20)² × (0.30)²

= 0.0216

Then the probability that these four will contain at least two more high-risk drivers than low-risk drivers is:

P (at least two more H than L) = P (HHHH) + P (HHHL) + P (HHHM)

+ P (HHMM)

= 0.0016 + 0.016 + 0.0096 + 0.0216

= 0.0488

Thus, the probability that these four will contain at least two more high-risk drivers than low-risk drivers is 0.0488.