Expected ValueDefinitionAmerican RouletteExample 2Two dice are rolledWho Wants to be a Millionaire?More MillionaireMathematical analysisWhat if we had a 50/50Expected ValueSection 3.5Definition•Let’s say that a game gives payoffs a1, a2,…, an with probabilities p1, p2,… pn. The expected value ( or expectation) E of this game is E = a1p1 + a2p2 + … + anpn.•Think of expected value as a long term average.American Roulette•At a roulette table in Las Vegas, you will find the following numbers 1 – 36, 0, 00. There are 38 total numbers.•Let’s say we play our favorite number, 7.•We place a $1 chip on 7. If the ball lands in the 7 slot we win $35 (net winnings). If the ball lands on any other number we lose our $1 chip.•What is the expectation of this bet?•To answer this question we need to know the probability of winning and losing.•The probability of winning is 1/38. The probability of losing is 37/38.•So the expectation is E = $35(1/38) + (-$1)(37/38) = (35-37)/38 = -2/38 = -$0.053•What this tells us is that over a long time for every $1 we bet we will lose $0.053.•This is an example of a game with a negative expectation. One should not play games when the expectation is negative.Example 2•On the basis of previous experience a librarian knows that the number of books checked out by a person visiting the library has the following probabilities:# of books 0 1 2 3 4 5Probability 0.15 0.35 0.25 0.15 0.05 0.05• Find the expected number of books checked out by a person.• E = 0(0.15)+1(0.35)+2(0.25)+3(0.15)+4(0.05)+5(0.05)• E = 0 + 0.35 + 0.50 + 0.45 + 0.20 + 0.25• E = 1.75Two dice are rolled•A player gets $5 if the two dice show the same number, or if the numbers on the dice are different then the player pays $1.•What is the expected value of this game?•What is the probability of winning $5? ANSWER 6/36 = 1/6.•What is the probability of paying a $1? ANSWER 5/6.•Thus E = $5(1/6) + (-$1)(5/6) = 5/6 – 5/6 = 0.•The Expectation is $0. This would be a fair game.Who Wants to be a Millionaire?•Recall the game show Who Wants to be a Millionaire? Hosted by Regis Philbin.•Let’s say you’re at the $125,000 question with no life-lines. The question you get is the following•What Philadelphia Eagles head coach has the most victories in franchise history? A. Earle “Greasy” Neale B. Buddy Ryan C. Dick Vermeil D. Andy ReidMore Millionaire•If you get the question right you will be at $125,000. •If you get the question wrong you fall back to $32,000.•Your third option is to walk away with $64,000.•What to do, what to do?•Let’s do a mathematical analysis.Mathematical analysis•What is the probability of guessing correctly? ANSWER ¼.•What is the probability of guessing incorrectly? ANSWER ¾.•What is our expectation? ANSWER E = $125,000(1/4)+$32,000(3/4) = $55,250. This is less than the $64,000 walk away value.•Decision: We should walk away.What if we had a 50/50•Then 2 of the choices would vanish.•Now the choices left will be A. Greasy Neale and D. Andy Reid.•Now E = (1/2)$125,000 + (1/2)$32,000 = $78,500 > $64,000.•We should give it a shot.•The answer is A. Greasy Neale (for now, later this season Andy Reid will pass
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