1
Would You Take This Bet?
• We flip a (fair) coin once.
– If it is heads, you win (at least) $2.
– If it is tails, you win nothing.
• We keep on flipping the coin.
– As long as the coin keeps landing on heads, your
winnings keep doubling:
• $4 on the second heads
• $8 on the third heads
• $16 on the fourth heads…
– We stop flipping on the first tails.
• In exchange, you owe me $1 million.
Random Variable
• A random variable is a set of outcomes plus a
probability associated with each outcome.
– The probabilities must sum to one.
• Example: coin flip
– Outcomes, probabilities: {Head, ½ ; Tail, ½ }
• Usually (but not necessarily) the outcomes are
numbers.
• Example: score on the midterm
– {100, probability = 1/4}
– {90, probability = 1/2}
– {80, probability = 1/4}
Expected Value
•
The average outcome of a draw from a random variable
(say, X).
•
The expected value equals the (probability) weighted
average of the outcomes.
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Examples of Expected Values
• Example: coin flip – let heads = 1, tails = 0
– E[coin flip] = 1 * ½ + 0 * ½ = ½
• Example: midterm grades:
– E[midterm grade] = 100 * ¼ + 90 * ½ + 80 * ¼ = 90
• Example: California Lottery
– “Half the money goes to the schools”
– For a $1 bet, the expected return is 50 cents.
St. Petersburg Paradox
• To calculate the expected value of the bet from the
beginning of the lecture, we need to find the
probability of a long series of heads uninterrupted
by a tails outcome:
– 1 heads; outcome = $2; probability = ½
– 2 heads; outcome = $4; probability = ½ * ½ = ¼
– 3 heads; outcome = $8; probability = ½ * ½ * ½ =
…
–
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Paradox (II)
• Your winnings in the bet are a random variable,
say W.
• The expected value of W is infinite:
• Yet I am charging only $1 million for the bet—it’s
a better deal than the California lottery.
• Paradox: no one will take this bet. Why?
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 Fall '07
 Tendall,M
 Utility, The Land, St. Petersburg paradox, Ms. Fogg

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