1. Suppose that an insurance company classifies people into...

1. Suppose that an insurance company classifies people into one of three classes: good risks, average risks and bad risks. The company's record indicates that the probabilities that good-, average- and bad-risks persons will be involved in an accident over a one-year span are, respectively, 0.1, 0.2 and 0.30. Suppose that 20 % of the population is a good risk, 50 %an average risk and 30 % percent a bad risk. a) What proportion of people have accidents in a fixed year? b) If policy holder A had no accidents over the last year, what is the probability that he or she is a bad risk?

Transcript

0:00 All right.

00:01 So the answer for a is 0 .21 and the answer for b is 0 .342.

00:12 Let's go over how we got this.

00:14 So what proportion of people have accidents in a fixed year? so i listed out the probability, probably a tree diagram here.

00:23 Point 1 is the probability that a good driver, good risk driver will have an accident.

00:29 Point 2 over average, 0 .3 for bad risk.

00:31 And then these are the proportions of people of the population, the percent of the population that's good risk, average risk in 30 percent, or excuse me, bad risk right here.

00:45 So the probability that an accident occurs, given they're a good driver, 0 .02, you multiply those probabilities together because you're following this path here, 0 .1 times 0 .2.

00:58 Probability there's, oh, and i'm saying e is accident.

01:02 E is an accident.

01:04 I didn't want to use a because we already have a for average.

01:16 Right, so there's that and probability that there's an accident if they're bad driver, 0 .310 .3, so you get 0 .09.

01:28 And then you add those up to get your probability of an accident.

01:31 So that's where the 0 .21 comes from.

01:34 Which means, and they have this over here, that means the probability of not having an accident is 0 .79, of picking someone at random and having them, or that's the proportion of people that have no accidents, the compliment.

01:54 So now the second part for erb, if a policy holder had no accidents over the last year, what's the probability that he or she has a bad risk? so this is what we're looking for.

02:04 The probability, given there's no accidents, what's the probability of their bad risk? that's equal to the following.

02:15 The probability of no accident given their risk times the probability of bad risk all over the probability of no accident right here.

02:27 So this, we know probability of b.

02:30 We know that already.

02:31 We know that's right here, 30%...

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