An auditor for Health Maintenance Services of Georgia...

An auditor for Health Maintenance Services of Georgia reports $40 \%$ of policyholders 55 years or older submit a claim during the year. Fifteen policyholders are randomly selected for company records. a. How many of the policyholders would you expect to have filed a claim within the last year? b. What is the probability that 10 of the selected policyholders submitted a claim last year? c. What is the probability that 10 or more of the selected policyholders submitted a claim last year? d. What is the probability that more than 10 of the selected policyholders submitted a claim last year?

An auditor for Health Maintenance Services of Georgia reports $40 \%$ of policyholders 55 years or older submit a claim during the year. Fifteen policyholders are randomly selected for company records.
a. How many of the policyholders would you expect to have filed a claim within the last year?
b. What is the probability that 10 of the selected policyholders submitted a claim last year?
c. What is the probability that 10 or more of the selected policyholders submitted a claim last year?
d. What is the probability that more than 10 of the selected policyholders submitted a claim last year?

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (often termed success and failure) and the probability of success is constant across trials.

Expected Value for a Binomial Random Variable

The expected value (or mean) of a binomial random variable is calculated by multiplying the number of trials by the probability of success in each trial. This value represents the average number of successes one would expect over many repetitions of the experiment.

Probability Mass Function (PMF) for the Binomial Distribution

The probability mass function for the binomial distribution gives the probability of achieving exactly a specific number of successes in a given number of trials. It is computed using the combination formula to account for the different ways the successes can occur, multiplied by the success probability raised to the number of successes, and the failure probability raised to the remaining number of trials.

Cumulative Probability and the Complement Rule

Cumulative probability refers to the probability that the binomial random variable is less than or equal to a certain value, which is found by summing the PMF over the desired range of outcomes. The complement rule can also be employed to calculate probabilities for ranges (such as 'greater than' or 'at least' a certain number) by subtracting the cumulative probability of the complementary range from one.

Transcript

00:01 All right, so we've got an auditor for the health maintenance service of georgia, 40 % of policyholders, 55 years or older, submit a claim during a year.

00:13 And this kind of tells you it's going to be a, this tells me it's going to be a binomial experiment because 40 % of some clearly defined probability, 40%.

00:29 And you could either be filing a claim or not filing it.

00:36 Sorry, we can already see this can be a binomial experiment.

00:40 And the effect of one person filing claim, we're assuming will not affect another person filing a claim.

00:50 So how many policyholders would we expect? so this is for the binomial, for a binomial distribution, the mean would be n, which is 50.

01:00 15 times the probability.

01:04 It says 15 policy holders are randomly selected.

01:08 So n is 15.

01:14 And then the probability is 0 .4.

01:18 So the mean is just n times p.

01:21 I think actually the text, this text uses pi.

01:25 So that's the way to, is the variable pie.

01:31 It's the probability, not the number pi.