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Casinos don’t gamble.

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Casinos don’t gamble.

Chapter 7C

The Law of Large Numbers

Expected Value

Gambler’s Fallacy

7-A

When tossing a fair coin, which of the following events is most likely?

A. Getting 8 tails in 10 tosses

B. Getting 80 tails in 100 tosses

C. Getting 800 tails in 1000 tosses

7-A

Applies to a process for which the probability of an event A is P(A) and the results of repeated trials are independent

If the process is repeated through many trials, the proportion of the trials in which event A occurs will be close to the probability P(A).

The larger the number of trials the closer the proportion should be to P(A).

7-A

This figure shows the results of a computer simulation of rolling a die.

The Law of Large Numbers

A roulette wheel has 37 numbers: 18 are black, 18 are red, and one green.

1. What is the probability of getting a red number on any spin?

2. If patrons in a casino spin the wheel 100,000 times, how many times should you expect a red number?

7-A

In a large casino, the house wins on its blackjack games with a probability of 50.7%. Which of the following events is the most likely? Most unlikely?

A. You win at a single game.

B. You come out ahead after playing forty times.

C. You win at a single game given that you have just won ten games in a row.

D. You win at a single game given that you have just lost ten games in a row.

7-A

7-A

Consider two events, each with its own value and probability.

Expected Value

= (event 1 value) • (event 1 probability)

+ (event 2 value) • (event 2 probability)

The formula can be extended to any number of events by including more terms in the sum.

7-A

You have been asked to play a dice game. You will roll one die and:

If you roll a 1, 2, 3 or 4, you will lose $30.

If you roll a 5, you will win $48

If you roll a 6, you will win $60

What is the expected value (to you) of this game?

7-A

Consider the following game. The game costs $1 to play and the payoffs are $5 for red, $3 for blue, $2 for yellow, and nothing for white. The following probabilities apply. What are your expected winnings? Does the game favor the player or the owner?

7-A

You are given 5 to 1 odds against tossing three heads with three coins.

This means that you win $5 if you succeed and you lose $1 if you fail.

Find the expected value (to you) of the game.

NOTE: A game is said to be "fair" if the expected value for winnings is 0, that is, in the long run, the player can expect to win 0.

7-A

Example: Suppose an automobile insurance company sells an insurance policy with an annual premium of $200. Based on data from past claims, the company has calculated the following probabilities:

An average of 1 in 50 policyholders will file a claim of $2,000

An average of 1 in 20 policyholders will file a claim of $1,000

An average of 1 in 10 policyholders will file a claim of $500

Assuming that the policyholder could file any of the claims above, what is the expected value to the company for each policy sold?

- The expected value is
- (This suggests that if the company sold many, many policies, on average, the return per policy is a positive $60.)

7-A

Let the $200 premium be positive (income) with a probability of 1 since there will be no policy without receipt of the premium. The insurance claims will be negatives (expenses).

7-A

An insurance company knows that the average cost to build a home in a new subdivision is $100,000, and that in any particular year, there is a 1 in 50 chance of a wildfire destroying all homes in the subdivision.

Based on these data, and assuming the insurance company wants a positive expected value when it sells policies, what is the minimum that the company must charge for fire insurance policies in this subdivision?

7-A

Textbook problems

Orange: pg448 #35, 38

Green: pg489 #45, 49

Homework Chapter 7C

Orange: Pg 445

Quick Quiz: 1, 2, 4, 5, 6

Exercises: 1, 2, 9, 15, 19, 31, 35, 38

Green: Pg 485

1, 2, 4, 5, 6, 11, 12, 19, 25, 29, 41, 45,49

In a family of six children, which sequence do you think is more likely? a. MFFMFMb. MMMMMMc. Both are equally likely