The House Edge:

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Chapter 20

• The House Edge
• Expected Values

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Thought Question 1
How much would you be willing to pay for a ticket
to a contest in which there was a 1 chance that
you would win 500 and a 99 chance you would win
nothing? Explain your answer.
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The Main Point...
Long-Term Gains, Losses and Expectations

• While we cannot predict individual outcomes, we
  can predict what happens (on average) in the long
  run.

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Long-Term Gains, Losses and Expectations

• Tickets to a school fund-raiser event sell for
  1.
• One ticket will be randomly chosen, the ticket
  owner receives 500.
• They expect to sell 1,000 tickets. Your ticket
  has a 1/1000 probability of winning.
• Two outcomes
• You win 500, net gain is 499.
• You do not win, net gain is -1.

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Expected Value

• Your expected gain (expected value) is
  (499)(0.001) (-1)(0.999) -0.50.
• long term, you lose an average of 0.50 each time
  (conceptually) you enter such a contest.
• Hey, the school needs to make a profit!

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Make a Decision, Which Do You Choose?
(1) A gift of 240, guaranteed. (2) A 25 chance
to win 1,000 and a 75 of getting nothing.

• First alternative EV240, no variation.
• Second alternative EV(1000)(0.25)
  (0)(0.75) 250
• Make a Decision

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Make a Decision, Which Do You Choose?
(1) A gift of 240, guaranteed. EV240 (2) A
25 chance to win 1,000 and a 75 of getting
nothing. EV250

• If choosing for ONE trial
• option (2) will maximize potential gain (1000)
• option (2) will minimize potential gain (0)
• option (1) guarantees a gain
• If choosing for MANY (500?) trials
• option (2) will maximize expected gain(will make
  more money in the long run)

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Make a Decision, Which Do You Choose?
(1) A sure loss of 740. (2) A 75 chance to lose
1,000 and a 25 to lose nothing.

• First alternative EV740, no variation.
• Second alternative EV(1000)(0.75)
  (0)(0.25) 750
• Make a Decision

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Make a Decision, Which Do You Choose?
(1) A sure loss of 740. EV740 (2) A 75
chance to lose 1,000 and a 25 to lose nothing.
EV750

• If choosing for ONE trial
• option (2) will minimize potential loss (0)
• option (2) will maximize potential loss (1000)
• option (1) guarantees a loss
• If choosing for MANY (500?) trials
• option (1) will minimize expected loss (will
  lose less money in the long run)

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The Law of Large Numbers

• The actual average (mean) outcome of many
  independent trials gets closer to the expected
  value as more trials are made.
• the higher the variability of the trials, the
  larger the sample needed
• expected values can be calculated by simulating
  many repetitions and finding the average of all
  of the outcomes

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The Law of Large NumbersGambling

• The house in a gambling operation is not
  gambling at all
• the games are defined so that the gambler has a
  negative expected gain per play
• each play is independent of previous plays, so
  the law of large numbers guarantees that the
  average winnings of a large number of customers
  will be close the the (negative) expected value
• State lottos have extremely variable outcomes
  also use pari-mutuel system for (fixed) payoffs
• law of large numbers does not apply

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Key Concepts

• Long-run Relative Frequency Interpretation of
  Probability
• Probability can be used to make accurate
  predictions about long-run averages and events.
• Law of Large Numbers