Section 12'4: Expected Value

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Section 12.4 Expected Value

• Dr. Fred Butler
• Math 121 Fall 2004

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Idea of Expected Value

• Expected value (also called expectation) is often
  used to determine the expected results of an
  experiment or business venture over the long run.
• It is used in business to predict future profits
  of a new product, in the insurance industry to
  determine how much each persons insurance rate
  is, and to predict the expected gain in games
  such as the lottery, roulette, craps, and slot
  machines.

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An Example of Expected Value

• Suppose I tell you I will give you 1 if you can
  roll an even number on a single die, and you have
  to give me 1 if you fail to roll an even number.
• Who would win money in the long run if this game
  were played many times?

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An Example of Expected Value contd.

• Since there are 3 even numbers (2, 4, 6) out of a
  total of 6 possible number (1,2,3,4,5,6), we can
  expect that about half the time you will win and
  I will give you 1, and about half the time you
  will lose and have to give me 1.
• Thus in the long run, we would probably both
  break even playing this game.

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Computing Expected Value Mathematically

• Mathematically we could find your expected gain
  or loss playing this game by the following
• P(You win)x(Amount you win)
• P(You lose)x(Amount you lost)
• (1/2)x(1)(1/2)x(-1) 1/2(-1/2) 0.
• Note that an amount of money that is lost is
  written as a negative number.
• Since the expected value of this game is 0, we
  call it a fair game.

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Formula for Expected Value

• In general, the expected value E is calculated by
  multiplying the probability of each possible
  event occurring by the net amount gained or lost
  if the event occurs, and adding these values
  together.
• If P1 is the probability that the first event
  occurs and A1 is the net amount won or lost if
  event 1 occurs, P2 is the probability that the
  second event occurs and A2 is the net amount won
  or lost if event 2 occurs, etc., then
• E P1xA1 P2xA2 PnxAn.

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Interpretation of Expected Value

• An expected value of zero indicates that you can
  expect to break even in the long run.
• A positive expected value indicates that you can
  expect to make money in the long run.
• A negative expected value indicates that you can
  expect to lose money in the long run.

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PRS Question 5.4

• Suppose you roll a single die and I tell you I
  will pay you 4 if you roll a 1, and you have to
  pay me 1 if you dont roll a 1. Can you expect
  to make money at this game in the long run?
• 1. Yes 2. No

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Answer to PRS Question 5.4

• P(Rolling 1)1/6 and P(Not rolling 1)1-1/65/6.
• Then
• E P(Rolling 1)x(Amount won)
• P(Not rolling 1)x(Amount lost)
• (1/6)x(4) (5/6)x(-1) (4-5)/6-1/6
• -0.17.
• So you can expect to lose around 17 cents at this
  game in the long run.

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Example 1

• Suppose you are taking a multiple choice exam in
  which each question has 5 possible answers.
• You will receive 2 points for each correct
  answer, but you will lost 1/2 point for each
  incorrect answer.
• Question If you dont know an answer, is it to
  your advantage to guess?

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Example 1 contd.

• If you completely guess on a question with 5
  choices, then
• P(Guess correctly)1/5
• P(Guess incorrectly)4/5.
• Then
• E P(Guess correctly)x(Points gained for
  correct answer)P(Guess incorrectly)x(Points lost
  for incorrect answer) (1/5)x(2)(4/5)x(-1/2)
  2/5 - 4/10 2/5 2/5 0.

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PRS Question 5.5

• Suppose you dont know the answer, but you can
  eliminate one of the choices as definitely not
  right. So there are 4 possible answers and you
  gain 2 points if you guess correctly, and lose
  1/2 point if you guess incorrectly.
• Is it to your advantage to guess in this case?
• 1. Yes 2. No

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Answer to PRS Question 5.5

• In this case
• P(Guess correctly)1/4
• P(Guess incorrectly)3/4
• and
• E (1/4)x(2)(3/4)x(-1/2) 2/4-3/8
• 4/8-3/81/8.
• In the long run you can expect to gain 1/8 point,
  so you should guess in this case.

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Example 2

• Suppose 1000 raffle tickets are sold for 1 each.
  There is one grand prize of 500, and two
  consolation prizes of 100 each. All of the
  tickets are placed in a bin and randomly selected
  for the prizes, and each ticket selected for a
  prize is returned to the bin before the next
  ticket is selected.
• Question What is the expected value if we
  purchase one ticket?

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Example 2 contd.

• There are three amounts to be considered
• 1. The net gain of winning the grand prize
• 2. The net gain of winning a consolation prize
• 3. The net loss of buying a ticket and winning
  nothing.
• The net gain in winning the grand prize is
  500-1499, the net gain in winning a
  consolation prize is 100-199, and the net
  loss or buying a ticket and winning nothing is
  -1.

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Example 2 contd.

• The probability of winning the grand prize is
  1/1000, the probability of winning a consolation
  prize is 2/1000, and the probability of winning
  nothing is
• 1-1/1000-2/1000997/1000.
• Thus
• E(1/1000)x(499)(2/1000)x(99)
• (997/1000)x(-1)
• (499198-997)/1000-300/1000-0.30.

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Example 3

• Question In the same raffle, what is our
  expected value if we buy five tickets?
• Our expected value when purchase one ticket is
  -0.30.
• Thus our expected value when purchasing five
  tickets is given by
• E 5x(Expected value when purchasing one
  ticket) 5x(-0.30) -1.50.

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Fair Price

• The fair price for playing a game is the amount
  to be paid that will result in an expected value
  of 0 (thus making it a fair game).
• Fair price can be found by the formula
• Fair price Expected value Cost to play.

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Fair Price Example

• For the example with 1000 raffle tickets sold for
  1 each, with one grand prize of 500 and two
  consolation prizes of 100 each, we found that
  E-0.30.
• Since the cost to play is 1, the formula tells
  us
• Fair price Expected value Cost to play
  -0.30 1 0.70.

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A Fair Price Example contd.

• So if the price for a ticket in this raffle were
  0.70, the expected value of this game would be
  0.
• We see directly that
• E (1/1000)(500-0.70) (2/1000)(100-0.70)
  (997/1000)(-0.70)
• (499.30198.60-697.90)/1000 0.
• So paying 0.70 for a raffle ticket (instead of
  1) makes the expectation 0, and makes this
  raffle a fair game.

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Example 4

• Suppose it costs 5 to play a game where you spin
  the pointer, and you win the amount that the
  pointer lands on.
• Question What is your expected value playing
  this game?
• We need to remember to subtract the cost to play
  from any prize you win.

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Example 4 contd.
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Example 4 contd.

• So you have a 1/2 chance of losing 4, a 1/4
  chance of losing 3, and a 1/4 chance of winning
  5.
• Thus E (1/2)(-4)(1/4)(-3)(1/4)(5)
• (-8-35)/4-6/4-1.50.
• So in the long run you can expect to lose 1.50
  playing this game.

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PRS Question 5.6

• What is a fair price for the spinner game?
• 1. 5 2. 3.50 3. 1.50

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Lecture Summary

• The expected value is computed by the formula
• E P1xA1 P2xA2 PnxAn.
• The fair price for playing a game is the amount
  to be paid that will result in an expected value
  of 0 .

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Homework

• Do problems from Section 12.4 of textbook.
• Have a nice Thanksgiving break!