EXPECTED VALUE

1
EXPECTED VALUE

• Suppose you and a friend are playing a game. You
  flip a coin.
• If the coin lands heads up, then your friend
  scores 1 point and you lose 1 point.
• If the coin lands tails up, then you score 1
  point and your friend loses
• 1 point.

2

• After playing the game many times, would you
  expect to have more, fewer, or the same number of
  points as when you started?
• The answer is that you should expect to end up
  with about the same number of points. In other
  words, you can expect to lose a point about 50
  of the time and win a point about 50 of the
  time.
• The expected value for this game is 0.

3

• A collection of outcomes is partitioned into n
  events, no two of which have any outcomes in
  common. The probabilities of n events occurring
  are p1 , p2 , p3 ,pn ,where p1 p2 p3 pn
  1.
• The values of the n events are x1 , x2 , x3, ,
  xn . The expected value V of the collection of
  outcomes is the sum of the products of the
  events probabilities and their values.
• V p1x1 p2x2 p3x3 pnxn

4

• You and a friend each flip a coin. If both coins
  land heads up, then your friend scores 3 points
  and you lose 3 points.
• If one or both of the coins land tails up, then
  you score 1 point and your friend loses 1 point.
• What is the expected value of the game from your
  point of view?

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Solution

• When the 2 coins are tossed, four outcomes are
  possible HH, HT, TH, and TT.
• Let event A be HH and event B be HT, TH, and TT.
• Note all possible outcomes are listed, but NO
  outcome is listed twice.
• The probabilities of the events are
• P(A) ¼ P(B) 3/4

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Solution Continued

• From your point of view the values of the events
  are
• Value of event A -3value of Event B1
• Therefore, the expected value of the game is
  V1/4 (-3) ¾ (1) 0
• What is a fair game? A fair game is one where
  each player has an equal chance of winning, and
  the expected value is 0.

7

• You participate in a game show in which you
  respond to questions that have 3 possible
  answers. You gain 10 for each correct answer and
  lose 6 for each incorrect answer. Every question
  must be answered.
• If you do not know the answer to one on the
  questions, is it to your advantage to guess the
  answer?

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Solution

• Step 1Find the probability of each outcome.
  Because each question has 1 right answer and 2
  wrong answers, the probability of guessing
  correctly is ___. The probability of guessing
  incorrectly is _____.
• Step 2Find the expected value of guessing an
  answer. Multiply the money gained or lost by the
  corresponding probability, then find the sum of
  these products.

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• Expected value
• _____ (1/3) ______ (2/3) _______
• Is is to your advantage to guess an answer? Why
  or why not?

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Finding an Expected Value in Real Life

• In 1996 there were 124,600,000 cars in use in the
  USA. That year there were 13,300,000 automobile
  accidents.
• The average premium paid in 1996 for automobile
  collision insurance was 685 per car, and the
  average auto collision claim paid by insurance
  companies was 2100 per car

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• What is the expected value of insurance coverage
  for a car with collision insurance in 1996?
• SOLUTION
• Let Event A be having an auto accident. Let Event
  B be NOT having an auto accident. Note that
  events A and B are mutually exclusive.
• The probabilities of the events are
• P(A) 13,300,000/124,600,000 0.107
• P(B) 1 P(A) 1 0.107 0.893

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• Here is how you calculate the values of the
  events
• If an insured car had an accident, the owner paid
  an average of 685 and received an average of
  2100.
• If a car did NOT have an accident, the owner only
  paid an average of 685.
• So, the values of the events for an insured
  person are
• value of event A -685 2100 1415
• Value of event B -685

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• Therefore, the expected value of insurance
  coverage was
• V 0.107(1415) 0.893(-685) -460
• Why is the expected value of the policy negative?
• Possible answer The expected value is negative
  because the insurance company has to pay overhead
  charges and earn some profit.
• How could they use the expected value to compare
  insurance coverage?
• The coverage with the highest expected value is
  the best deal, although there may be other
  reasons for preferring one policy over another.

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Practice Problem

• A movie theater is giving away a 100 prize and a
  50 prize.
• To enter the drawing, you need to buy a movie
  ticket for 6. The ticket collectors will take
  the tickets from the first 1000 guests, and after
  the movie ends, they will randomly choose one
  ticket. If the number chosen matches the number
  on your ticket stub, you win 1st or 2nd prize.
  What is the expected value of your gain?

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• Step 1 Find the gain for each prize by
  subtracting the cost of the ticket from the prize
  money.
• Step 2 Find the probability of each outcome.
  There are 1000 tickets sold and the probability
  of winning one of the prizes is ________. Because
  there are two prizes, there are 2 winning tickets
  and _____ losing tickets. So, the probability you
  will NOT win a prize is __________.

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• Step 3Summarize the information
• Gain, x _____ _____ _____
• Probability 1/1000 1/1000 998/1000
• Step 4Find the expected value by finding the sum
  of each outcome multiplied by its corresponding
  probability.
• V ___ (1/1000) ____(1/1000) ___(998/1000)
  _______
• The expected value of your gain is _____. This
  means that you can expect to _____
• an average of ______ for each ticket you buy.