You Bet Your Life - So to Speak Chapter 16

Life Insurance An insurance company offers a “death and disability” policy that pays $10,000 when you die or $5000 if you are permanently disabled. It charges a premium of only $50 a year for this benefit. Is the company likely to make a profit selling such a plan? The company looks at the probability that its clients will die or be disabled in any year. This actuarial information helps the company calculate the expected value of this policy.

We’ll want to build a probability model in order to answer the questions about the insurance company’s risk. First we need to define a few terms.

Random Variables Random variable - a numeric value which is based on the outcome of a random event. Represent with a capital letter, like X Use the lowercase to denote any particular value the Random Variable have, like x Note: The most common letters are X, Y, and Z. But be cautious: If you see any capital letter, it just might denote a random variable.

Random Variables For the insurance company, x can be $10,000 (if you die that year) $5000 (if you are disabled) $0 (if neither occurs).

Probability Model The Probability Model for the Random Variable is the collection of all the possible values and their probabilities.

Suppose, the death rate in any year is 1 out of every 1000 people, and that another 2 out of 1000 suffer some kind of disability. Then we can display the probability model for this insurance policy in a table like this:

Expected Value (Center) The expected value is a parameter In fact, it’s the mean. We’ll signify this with the notation (for population mean) or E(X) for expected value.

What Can the Insurance Company Expect? Imagine the company insures exactly 1000 people. Further imagine that, in a perfect “probability world,”: 1 of the policyholders dies - $10,000 2 are disabled - $5000 each 997 survive the year unscathed - $0 each Since it is charging people $50 for the policy, the company expects to make a profit of $30 per customer.

We can’t predict what will happen during any given year, but we can say what we expect to happen. To do this, we (or, rather, the insurance company) need the probability model. How did we come up with $20 as the expected value of a policy payout? We imagined that we had exactly 1000 clients. Of those, we imagined exactly 1 died and 2 were disabled, corresponding to the probabilities. Our average payout is:

Expected Value

A $20 bill, two $10 bills, three $5 bills and four $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen? Outcome Probability $20 1/10 $10 2/10 $5 3/10 $1 4/10 The expected value is $5.90

In a game you flip a coin twice, and record the number of heads that occur. You get 10 points for 2 heads, zero points for 1 head, and 5 points for no heads. What is the expected value for the number of points you’ll win per turn? Outcome Probability 2 Heads 1/4 1 Head 1/2 No Heads The expected value is 3.75

There is an equally likely chance that a falling dart will land anywhere on the rug below. The following system is used to find the number of points the player wins. What is the expected value for the number of points won? Black = 40 points Gray = 20 points White = 0 points

Outcome Value Probability Black 40 6/15 = 2/5 Gray 20 White 3/15 = 1/5

A mysterious card-playing squirrel (pictured) offers you the opportunity to join in his game. The rules are: To play you must pay him $2. If you pick a spade from a shuffled pack, you win $9. Find the expected value you win (or lose) per game. Outcome Value Probability Spade $9- $2 = $7 1/4 Other -$2 3/4

A dice game involves rolling 2 dice A dice game involves rolling 2 dice. If you roll a 2, 3, 4, 10, 11, or a 12 you win $5. If you roll a 5, 6, 7, 8, or 9 you lose $5. Find the expected value you win (or lose) per game. Outcome (Sum) Value Probability 2, 3, 4, 10, 11, 12 $5 6/11 5, 6, 7, 8, 9 -$5 5/11

Recall: Population Mean = Expected Value

First Center, Now Spread . . . Of course, this expected value (or mean) is not what actually happens to any particular policyholder. No individual policy actually costs the company $20. We are dealing with random events, so some policyholders receive big payouts, others nothing. Because the insurance company must anticipate this variability, it needs to know the standard deviation of the random variable. For data, we calculated the standard deviation by first computing the deviation from the mean and squaring it. We do that with (discrete) random variables as well.

First, we find the deviation of each payout from the mean (expected value):

Skills

Classwork & Homework Classwork: 16.1 WS Probability Homework: pp. 363 – 364 (1 – 8)