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.
The Law of Large Numbers
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
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).
This figure shows the results of a computer simulation of rolling a die.
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?
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.
Consider two events, each with its own value and probability.
= (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.
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?
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?
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.
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?
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).
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?
Orange: pg448 #35, 38
Green: pg489 #45, 49
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