It is known that about 20% of car insurance policies include emergency road service. Suppose we randomly select 10 car insurance policies.

What is the probability that at least one policy will have emergency road service?
Answer

0.6242

0.3758

0.8926

0.2684

0.1074

What is the expected number of policies having emergency road service?
Answer

3.5

8.0

1.6

2.0

5.0

I couldn't figure out the second part. for the first one i got .3758

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How did you get 0.3748 for part 1? The correct answer is 1 - (0.8)^10

One minus the probability that none of the ten have roadside coverage

The expecation value is the single-policy probabiity x (number of policies). That product is 2.

To calculate the probability that at least one policy will have emergency road service, we first need to find the probability that none of the selected policies have emergency road service.

Since each policy has a 20% chance of having emergency road service, the probability of a single policy not having it is 1 - 0.20 = 0.80.

To find the probability of none of the 10 policies having emergency road service, we need to multiply the probability of each policy not having it:

0.80 * 0.80 * 0.80 * 0.80 * 0.80 * 0.80 * 0.80 * 0.80 * 0.80 * 0.80 ≈ 0.1074

Therefore, the probability that at least one policy will have emergency road service is 1 - 0.1074 = 0.8926. Thus, the correct answer is 0.8926.

For the second part of the question, we can use a similar approach. The expected value, or mean, of a random variable is calculated by multiplying each possible outcome by its respective probability and summing them up.

In this case, the random variable represents the number of policies with emergency road service among the 10 selected policies. The possible outcomes range from 0 (none of the policies have emergency road service) to 10 (all of the policies have emergency road service).

We can calculate the expected value by multiplying each outcome by its probability and summing them up:

0 * (probability of 0 policies with emergency road service) +
1 * (probability of 1 policy with emergency road service) +
2 * (probability of 2 policies with emergency road service) +
3 * (probability of 3 policies with emergency road service) +
... +
10 * (probability of 10 policies with emergency road service)

To find the expected number of policies having emergency road service, we multiply the number of selected policies (10) by the probability of a single policy having emergency road service (0.20):

10 * 0.20 = 2.0

Therefore, the expected number of policies having emergency road service is 2.0. Thus, the correct answer is 2.0.