**Answer:**

**a) **0.999 = 99.9% probability that at least one had earthquake insurance

**b) **0.892 = 89.2% probability that four or more have earthquake insurance

**Step-by-step explanation:**

For each homeowner, there are only two possible outcomes. Either they have invested in earthquake insurance, or they have not. The probability of a home owner having invested in earthquake insurance is independent from other homeowners. So we use the binomial probability distribution to solve this question.

**Binomial probability distribution**

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And p is the probability of X happening.

**Suppose that 3 out of every 10 homeowners in the state of California has invested in earthquake insurance.**

This means that

**Suppose 20 homeowners are randomly chosen to be interviewed.**

This means that .

**(a) What is the probability that at least one had earthquake insurance? (Round your answer to three decimal places.)**

Either none had earthquake insurance, or at least one did. The sum of the probabilities of these events is 1. So

We want . So

In which

So

0.999 = 99.9% probability that at least one had earthquake insurance

**(b) What is the probability that four or more have earthquake insurance? (Round your answer to three decimal places.)**

Either less than 4 had earthquake insurance, or at least four did. The sum of the probabilities of these events is 1. So

We want . So

In which

So

0.892 = 89.2% probability that four or more have earthquake insurance