# Expected Value. 24 February Expected Value 24 February /19

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1 Expected Value 24 February 2014 Expected Value 24 February /19

2 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery games. We looked at some probability calculations about poker last week. But, that is only part of the story. How much you would win on a given hand is also important. Expected value involves both probability and payout. We ll start today with a video of playing a poker hand. Expected Value 24 February /19

3 Pot Odds Suppose you are playing heads up with one other player, and it is down to the last bet. Suppose there is \$100 in the pot and your opponent has made a \$10 bet. You are wondering if you should call the bet or fold. You estimate you have a 5 : 1 chance of wining the hand. What do you do? What you do is calculate pot odds. This is the ratio of pot size to bet. In this example it is 100 : 10, or 10 : 1. If your odds of winning are better than 10 : 1, which is the case here, it is to your advantage to call the bet. Expected Value 24 February /19

4 In the video clip we saw there was 135 on the table when Data folded. He had to bet 10 to stay in. Q Did Data make a good decision? Enter A for yes, enter B for no. The pot odds were 135 : 10, or 13.5 : 1. He should have bet if he had a 13 : 1 or better chance of winning. Did he? He would have won as long as Riker didn t have a heart. A Data blew it! His chance of winning was way better than 1 out of 13 times. Expected Value 24 February /19

5 One way the idea of pot odds is used is to decide it if it worthwhile to bet or to fold. If the possible payout isn t enough to warrant the bet, given the odds, folding is a good idea. Another way it is used is to try to get people to make bad bets. By betting enough so that your opponent s pot odds won t be good enough to warrant betting, you will either get them to fold or to make a bad bet. This is a sophisticated play, but very good players use this regularly. Expected Value 24 February /19

6 Expected Value Knowing the probability of winning a game isn t enough to know whether or not it is a good game to play. You also need to take into account how much you win when you win. Another aspect of this idea is how are payouts determined by casinos. Roughly, if you bet on the outcome of a game, such as the super bowl, the payout is related to the estimated probability that a given team will win. The more likely a team is thought to win, the lower the payout. In order to use this idea, you must be able to compute if you re more likely to make money than lose money with a given action. This leads us to the idea of expected value. Expected Value 24 February /19

7 We can quantify this idea. The resulting term is called Expected Value. The expected value of a bet is the amount of money one expects to win (or lose) when the bet is made many many times. For example, suppose you bet \$1 on whether the flip of a coin comes up heads. You win \$1 if the coin comes up heads. You lose your bet if it comes up heads. If you play this game many many times, the most likely outcome is to break even; you expect to win 50% of the time and lose 50% of the time. Since you win or lose \$1 each time you play, neither person playing has an advantage. Your expected value, or expected amount of winnings, is then 0 for each bet. Expected Value 24 February /19

8 Let s look at another example. Suppose we spin the following spinner. Suppose you play a game where you win if the spinner lands on blue and you lose otherwise. For a \$1 bet, if you win you receive \$3. Is this a good game for you to play? The probability of winning is 1/5. That means, on average, for every 5 times you play you win once. So, on average, every 5 times you play, you win 1 time and lose 4 times. If you bet \$1 each time you ll win \$3 and lose \$4, on average. So, you will lose \$1 out of every 5 bets, on average, or \$0.20 per bet. Expected Value 24 February /19

9 In other words, the expected value of playing is \$0.20 per \$1 bet. It is negative because it is more likely for you to lose than to win. Note the phrase on average. You could play and win 100 straight times. That isn t likely, but it is possible. Expected value indicates what is likely to happen in the long run, when you play many many times. It may not be representative of a day s worth of playing. But, the more you play the more likely the expected value represents what happens. For example, a casino can estimate fairly well how much money they ll make, given the number of people playing, by knowing the probability of winning and their payouts. This is because they have a huge number of players. Expected Value 24 February /19

10 A Formula for Expected Value In general, if you make a bet, then, on average, the amount of money you win (or lose) is given by Expected Value = probability of winning amount won probability of losing bet amount For example, in the game mentioned on the last slide, this would give \$3 (1/5) + ( \$1) 4/5 = \$0.20 Expected Value 24 February /19

11 Roulette American roulette is played on a roulette wheel with 38 slots. There are 18 black, 18 red, and 2 green (0 and 00). Expected Value 24 February /19

12 European roulette does not have a 00. Let s see a short clip of roulette playing. I suspect this clip isn t too representative of what actually happens in a casino. Expected Value 24 February /19

13 The Betting Board There are several different bets. For example, Bet on a single number. The payout is \$35 for a \$1 bet. Bet on red or black. The payout is \$1 for a \$1 bet. Bet on even or odd. The payout is \$1 for a \$1 bet. Bet on first, second, or third 12. The payout is \$2 for a \$1 bet. Expected Value 24 February /19

14 Probability of Winning Various Roulette Bets There are 38 equally likely outcomes in roulette, the 36 numbers 1 through 36, 0, and 00. If we bet on a single number, then there is exactly 1 way to win. The probability of wining is then 1 38 The probability of losing is then Expected Value 24 February /19

15 Expected Value of Betting on a Number What is the expected value of betting \$1 on a single number? We use the formula for expected value we gave earlier: Expected Value = probability of winning amount won probability of losing bet amount Our bet is \$1 and our payout is \$35. The expected value is then \$35 \$1 = = 2 38 which is about negative 5. Expected Value 24 February /19

16 Expected Value of Betting on a Color Q What is the expected value of betting on red? Recall the payout for a \$1 bet is \$1. Don t enter the negative sign. A There are 18 red numbers out of 38, so the probability of winning is 18/38 and the probability of losing is 20/38.The expected value is then \$ \$1 = 2 38 which is the same as for betting on a number. Expected Value 24 February /19

17 Expected Value of Betting on a Third Q What is the expected value of betting on the first 12? The payout for a \$1 bet is \$2. Again, don t enter the negative sign. A There rare 12 winning numbers (1-12) and 26 losing numbers. The probability of winning is then 12/38 and the probability of losing is 26/38. The expected value is then \$2 \$1 = = 2 38 Again, this is the same as the previous expected values. Expected Value 24 February /19

18 Next Time We will look at the New Mexico Lottery games, compute probabilities of winning and determine the expected value of different games. We ll see that, even for the same game, not all variants of playing have the same expected value. Expected Value 24 February /19

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