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College Algebra Sixth Edition James Stewart Lothar Redlin Saleem Watson 9 Probability and Statistics 9.4 Expected Value Expected Value Suppose that a coin has probability 0.8 of showing heads. • If the coin is tossed many times, we would expect to get heads about 80% of the time. • Now, suppose that you get a payout of one dollar for each head. • If you play this game many times, you would expect on average to gain $0.80 per game Expected Value—Definition A game gives payoff a1, a2, … , an with probabilities p1, p2, … , pn. The expected value (or expectation) E of this game is E = a1p1 + a2p2 + … + an pn E.g. 1—Finding Expected Value A die is rolled. • You receive $1 for each point that shows. • What is your expectation? E.g. 1—Finding Expected Value Each face of the die has probability 1/6 of showing. • So, you get $1 with probability 1/6, $2 with probability 1/6, $3 with probability 1/6, and so on. • Thus, the expected value is 1 1 1 1 1 1 E 1 2 3 4 5 6 6 6 6 6 6 6 21 3.5 6 E.g. 1—Finding Expected Value So if you play this game many times, you will make, on average, $3.50 per game. E.g. 2—Finding Expected Value In Monte Carlo, the game of roulette is played on a wheel with slots numbered 0, 1, 2, …, 36. • The wheel is spun, and a ball dropped in the wheel is equally likely to end up in any one of the slots. • To play the game, you bet $1 on any number other than zero. • For example, you may bet $1 on number 23. E.g. 2—Finding Expected Value If the ball stops in your slot, you get $36. • The $1 you bet plus $35. • Find the expected value of this game. E.g. 2—Finding Expected Value You gain $35 with probability 1/37, and you lose $1 with probability 36/37. • Thus, 1 36 E 35 1 37 37 0.027 E.g. 2—Finding Expected Value In other words, if you play this game many times, you would expect to lose 2.7 cents on every dollar you bet (on average). • Consequently, the house expects to gain 2.7 cents on every dollar you bet.