Notes 7C - Chandler Unified School District



Notes 7C The Law of Large Numbers

Ex 1) Roulette

A roulette wheel has 38 numbers: 18 are black, 18 are red, and the numbers 0 and 00 are green. Total = 18 + 18 + 2 = 38

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

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

Ex 2) Lottery Expectations

Suppose that $1 lottery tickets have the following probabilities: 1 in 5 to win a free ticket (worth $1), 1 in 100 to win $5, 1 in 100,000 to win $1000, and 1 in 10 million to win $1 million. What is the expected value of a lottery ticket? Discuss the implications. (Note: Winners do not get back the $1 they spend on the ticket.)

|Event |Value |Probability |Value [pic] Probability |

|Ticket purchase | | | |

|Win free ticket | | | |

|Win $5 | | | |

|Win $1000 | | | |

|Win $1 million | | | |

Sum:

Averaged over many tickets, you should expect to lose _______ for each $1 ticket that you buy!!

Ex 3) Art Auction

You are at an art auction, trying to decide whether to bid $50,000 for a particular painting. You believe that you have ½ probability of reselling the painting to a client in NY for $70,000 and a ¼ probability of reselling the painting to a client in San Francisco for $80,000. Otherwise, you will have to keep the painting –an option that has benefits, but no monetary ones. What is the expected value of a $50,000 bid?

Conclusion:

The GAMBLER’S FALLACY is the mistaken belief that a streak of bad luck makes a person “due” for a streak of good luck.

Ex 4) Continued Losses

You are playing the coin toss game in which you win $1 for heads and lose $1 for tails. After 100 tosses, you are $10 in the hole because you have 45 heads and 55 tails. You continue playing until you’ve tossed the coin 1000 times, at which point you have gotten 480 heads and 520 tails. Does the result agree with what we expect from the law of large numbers? Have you gained back any of your losses? Explain.

Ex 5) Hot Hand at the Craps Table

The popular casino game of craps involves rolling dice. Suppose you are playing craps and suddenly find yourself with a “hot hand”: You roll a winner on ten consecutive bets. Is your hand really “hot”? Should you increase your bet because you are on a hot streak? Assume that you are making bets with a 0.486 probability of winning on a single play.

Moral of this lesson:

THE HOUSE EDGE

A casino makes money because games are set up so that the expected earnings of patrons are negative (that is, losses). Because the casino earns whatever patrons lose, the casino’s earnings are positive. The amount that the casino, or house, can expect to earn per dollar bet is called the house edge. That is, the expected value to the casino of a particular bet.

Ex 6) The House Edge in Roulette

The game of roulette is usually set up so that betting on red is a 1 to 1 bet. That is, you win the same amount of money as you bet if red comes up. Betting on a single number is a 35 to 1 bet. That is, you win 35 times as much as you bet if your number comes up. What is the house edge in each of these two cases? If patrons wager $1 million on such bets, how much should the casino expect to earn?

Remember: In this problem you see yourself as the owner of the casino not as the gambler!!

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The Law of Large Numbers applies to a process for which the probability of an event A is P(A) and the results of repeated trials are independent. It states:

If the process is repeated over 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).

EXPECTED VALUE

Refers to expected gain or loss over many purchases.

Consider two events, each with its own value and probability. The expected value is:

[pic][pic]+ …

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

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