lecture15 - Would You Take This Bet? We flip a (fair) coin...

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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 • $8onthethirdheads • $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. [] å = = n i i i x X E 1 π n n x x x X y probabilit with ... y probabilit with y probabilit with ... 2 1 2 1 ï î ï í ì = Examples of Expected Values • Example: coin flip – let heads = 1, tails = 0 – E[coinfl ip]=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 = n heads; outcome = $2 n ; probability = 8 1 n ÷ ø ö ç è æ 2 1 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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lecture15 - Would You Take This Bet? We flip a (fair) coin...

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