Statistics Concepts and Controversies

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Statistics Concepts and Controversies

• Chapter 20
• The House Edge Expected Values

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Expected Values

• Expected values combine outcomes with their
  probabilities to give long-term or average
  outcomes of a numerical random phenomenon.
• An expected value is an average of the possible
  outcomes that weighs each outcome by its
  probability.
• Outcomes that occur more often get higher
  weights.
• Outcomes that occur less often get lower weights.
• Expected values relate to the big picture, not
  the individual outcomes.
• Gambling Example A gambler cares about how much
  he/she will win in the long run in addition to
  his/her probability of winning.

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Calculating Expected Values
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Tri-State Lottery Example 1

• Tri-State (New Hampshire, Maine, and Vermont)
  Lottery
• Pay 1 and choose a 3 digit number (i.e., 123).
• The state chooses a 3 digit number at random and
  pays you 500 if your number is chosen.
• There are 1000 3 digit numbers.
• Your probability of winning is 1/1000 (0.001).

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Expected Winnings (Long-run Average) in the
Tri-State Lottery

• The long run average winnings from a ticket are
  0.50.
• Because each ticket costs 1, in the long run you
  lose 50 cents on each lottery ticket.

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Rolling the Die Example

• What is the expected value for the roll of a fair
  die?
• What if the die has two 4s and no 1?

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Multiple Choice Test Example

• Some tests penalize for guessing. If there are 5
  choices then they will give 5 points for a right
  answer, 1 point for no answer, and 0 points for a
  wrong answer.
• What is the expected number of points for a
  guess?
• What if you can eliminate one answer as wrong?

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The Law of Large Numbers

• An expected value is an average of the possible
  outcomes, which outcomes with a higher
  probability count more (weighted higher).
• An expected value also represents the long-run
  average we will actually see if we place a bet
  many times or choose many households at random.
• An expected value is the long-run average.
• The Law of Large Numbers
• If a random phenomenon with numerical outcomes is
  repeated many times independently, the mean of
  the actually observed outcomes approaches the
  expected value.

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The Law of Large Numbers

• The law of large numbers is closely related to
  the idea of probability.
• In many independent repetitions, the proportion
  of each possible outcome will be close to its
  probability.
• House advantage Casinos are guaranteed a profit
  if enough bets are placed in a casino.
• Insurance companies
• Insurance companies know the probabilities and
  rely on the law of large numbers to predict the
  average amount (expected value) they will have to
  pay out.
• Then the companies set premiums high enough to
  guarantee a profit.

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Thinking About Expected Values

• How large is a large number?
• Depends on the variability of the random
  outcomes.
• The more variable outcomes, the more trials are
  needed to ensure that the mean outcome is close
  to the expected value.
• Is there a winning system to beat the odds?
• Does doubling your bet on the roulette wheel beat
  the odds?
• Such a system tries to take advantage of the fact
  that you have a memory even though the roulette
  wheel does not.
• No. If you do not have an infinite fortune to
  gamble, your average winnings (expected value)
  remains the same as long as successive trials of
  the game are independent (such as spins on the
  roulette wheel).

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Roulette Wheel Example
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Roulette Example
What is your expected winnings (average winnings)
if you play roulette until you win (max 4 times
in a row)?
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Simulating Expected Values

• You have a 30 chance of making a basket when
  throwing a wadded up paper into the trash. If you
  miss the first time, your second shot is made
  from closer and you have a 70 chance of making
  it.
• If you miss again you will just drop it in from
  next to the can (100).
• On average, how many shots will it take?
• Recognize that this is an expected value
  question.
• Using the random number table
• Shot 1
• 0-2 is a basket (30 chance)
• 3-9 is a miss (70 chance)
• Shot 2
• 0-6 is a basket (70)
• 7-9 is a miss (30)

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Simulating Expected Values
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Another Example

• The World Health Organization has some money that
  it can invest in either a new experimental
  vaccine or in new water systems in developing
  countries.
• If the vaccine works it will save 1,000,000 lives
  
• If it does not work then it will not save any
  lives.
• The new water systems are known to work and will
  save 500,000 lives.
• If we estimate that the vaccine has a 40 chance
  of working, what is the expected number of lives
  it will save?
• 1,000, 000 x 0.4 400,000
• What if it has a 70 chance of working?
• 1,000,000 x 0.7 700,000

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Statistics Concepts and Controversies
That's All Folks

• Chapter 20
• The House Edge Expected Values