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1 Expected Value CSCE 115 Revised Nov. 29, 2004, May 2, 2005 2 Probability Probability is determination of the chances of picking a particular sample from a known sample. Notation: if A is some event, the probability of A is P(A) 3 Probability Probability of success (when the events are equally likely) is Number of successful outcomes Number of possible outcomes Example: If 1 student is picked at random from a class of 7 woman and 13 men, what is the probability that the student is a woman? P(woman) = 7/(13+7) = 7/20 4 Probability Non-example: If you roll two dice and add the spots the possible outcomes are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. What is the probability of rolling a 2? Is the following correct? There is 1 success (getting a two) out of 11 possibilities (2, 3, …, 12) so the probability is 1/11 Why? 5 Independent events Two events are independent if the way the first event happens does not affect the way the second event happens. 6 Example: Independent events Put 3 red balls and 2 green balls in a bag. Event 1: select a ball at random from the bag and determine its color. Put the ball back. Event 2: Select a second ball at random. Events 1 and 2 are ____________ Event 3: select a ball at random and set it aside. Event 4: select a second ball at random. Events 3 and 4 are ____ _____________ 7 First fundamental rule: The probability that something does not happens is 1 - the probability it happens P(not A) = 1 - P(A) Example: The probability of picking a man from the class of 7 women and 13 men is 1 - (7/20) = 20/20 - (7/20) = 13/20 8 Second fundamental rule Iftwo events are independent, the probability that both A and B happen is P(A and B) = P(A) * P(B) Example: We randomly select a ball from a bag with 3 red and 2 green balls. We put it back and draw again. The probability that both balls are red is P(red, red) = (3/5) * (3/5) = 9/25 9 Example: Role 2 dice Suppose that we have a red die and a blue die. We roll and sum. What are the possible outcomes? RB RB RB RB RB RB 1,1 1,2 1,3 1,4 1,5 1,6 2,1 2,2 2,3 2,4 2,5 2,6 3,1 3,2 3,3 3,4 3,5 3,6 4,1 4,2 4.3 4,4 4,5 4,6 5,1 5,2 5,3 5,4 5,5 5,6 6,1 6,2 6,3 6,4 6,5 6,6 10 Example: Role 2 dice P(2) = 1/36 P(8) = 5/36 P(3) = 2/36 = 1/18 P(9) = 4/36 = 1/9 P(4) = 3/36 = 1/12 P(10) = 3/36 = 1/12 P(5) = 4/36 = 1/9 P(11) = 2/36 = 1/18 P(6) = 5/36 P(12) = 1/36 P(7) = 6/36 = 1/6 11 Probability of rolling 2 dice 0.2 Probability. 0.15 0.1 0.05 0 2 3 4 5 6 7 8 9 10 11 12 Sum of two dice 12 A children's game with spinner The spinner is used to 3 3 determine how far you 7 5 move. What is the probability of each 3 2 move? P(2) = 1/10 8 6 P(3) = 4/10 3 6 P(5) = 1/10 How far, on the average, P(6) = 2/10 P(7) = 1/10 do you expect to move P(8) = 1/10 each time you spin? 13 A children's game with spinner Just averaging the possible the possible outcomes (2 + 3 + 5 + 6 + 7 + 8) = 31 = 5.1667 6 6 is _____ correct because the various values are not equally likely. A correct way is (2 + 3 + 3 + 3 + 3 + 5 + 6 + 6 + 7 + 8) = 46 10 10 = 4.6 14 A children's game with spinner (2 +3 + 3 + 3 + 3 + 5 + 6 + 6 + 7 + 8) 10 = (2 +3*4 + 5 + 6*2 + 7 + 8) 10 = 2 * 1 + 3 * 4 + 5 * 1 + 6 * 2 + 7 * 1 + 8 * 1 10 10 10 10 10 10 = 2*P(2) + 3*P(3) + 5*P(5) + 6*P(6) + 7*P(7) + 8 * P(8) 15 Expected value Suppose that a certain experiment X could result in the values of {a, b, c, …, k} and the probabilities of these outcomes are P(a), P(b), P(c), …, P(k). The expected value is E(X) = a * P(a) + b * P(b) + c * P(c) + … + k * P(k) 16 Example: Expected value Recall the kid's spinner game P(2) = 1/10 P(3) = 4/10 P(5) = 1/10 P(6) = 2/10 P(7) = 1/10 P(8)= 1/10 E(spin) = 2 * .1 + 3 * .4 + 5 * .1 + 6 * .2 + 7 * .1 + 8 * .1 = 4.6 17 Example: Expected value Experiment: You flip a coin and get 1 point of a head and 0 points for tail P(head) = .5, P(tail) = .5 E(flip) = 1 * .5 + 0 * .5 = .5 18 Example: Expected value Experiment: You roll one die and count dots. P*(1) = 1/6, P(2) = 1/6, … P(6) = 1/6 E(roll) = 1*(1/6) + 2*(1/6) + 3*(1/6) + 4*(1/6) + 5*(1/6) +6 *(1/6) = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 = 21/6 = 3.5 19 Example: Expected value Experiment: You roll two dice and count dots P(2) = 1/36 P(6) = 5/36 P(9) = 4/36 P(3) = 2/36 P(7) = 6/36 P(10) = 3/36 P(4) = 3/36 P(8) = 5/36 P(11) = 2/36 P(5) = 4/36 P(12) = 1/36 E(two dice) = 2*(1/36) + 3*(2/36) + 4*(3/36) + 5*(4/36) 6*(5/36) + 7*(6/36) + 8*(5/36) + 9*(4/36) 10*(3/36) + 11*(2/36) + 12*(1/36) = 252/36 = 7 20 Example: Expected value John proposes that the charity bazaar sell tickets for $2. The player rolls 2 dice. The play wins $12 if the roll is a 2 or a 8. On the average, how much will the charity win each time a player rolls the dice? 21 Example: Charity bazaar Solution 1: Outcomes are P(2) = 1/36, Prize(2) = $12 P(8) = 5/36, Prize(8) = $12 P(other) = 30/36, Prize(other) = $0 E(Prize) = $12*(1/36) + $12*(5/36) + $0*(30/36) = $72/36 = $2 Cost of ticket - Expected value of prize = $2 - $2 = 0 The charity does not expect to win any money with this game. 22 Example: Charity bazaar Solution 2: Outcomes are charity wins $2 or loses $10 P($2) = 30/36 = 5/6 P(-$10) = 1/36 + 5/36 = 6/36 = 1/6 E(winnings) = $2 * (5/6) + (-$10) * (1/6) $10/6 + (-$10/6) = 0 23 Fair Game A fair game is a game where the expected value of winning is 0 Fair games are highly desirable when play with your friends Fair games are not desirable for organizations trying to earn money by offering games of chance. Casinos would go out of business if they had fair games. 24 Example: Charity bazaar John proposes that the charity bazaar sell tickets for $2. The player rolls 2 dice. The player wins $12 if the roll is a 2 or a 11. They win nothing otherwise. On the average, how much will the charity win each time a player rolls the dice? P(-$10) = 1/36 + 2/36 = 3/36 = 1/12 P($2) = 1- 1/12 = 11/12 E(winnings) = (-$10) * (1/12) + ($2)*(11/12) = (-$10 + $22)/12 = $12/12 = $1 The game is ___ fair. This is desirable for _____. 25 Odds Unfortunately popular slang uses “odds” in at least 3 different ways. It may indicate the payoff if you win a bet It may be a synonym for probability – This is used by the Washington State Lottery – It is used in many popular articles about odds may indicate the ratio of the probability of It winning to the probability of losing – This is the definition we will use – This the definition is the one normally sees in math and statistics books 26 Odds of Winning Suppose the probability of winning is p and probability of losing is q = 1 - p. Then the odds of winning are p:q. We treat p:q as a fraction and normally multiply and divide both parts to clear of fractions and to remove common factors. The odds of losing are q:p. 27 Example: Odds In therevised charity bazaar game, the probability of a player winning is 1/12. The probability of losing is ____ The odds of winning are 1/12:11/12 or ____ The odds of losing are ______ 28 Example: Odds You rolltwo dice and win if you roll a 9. The probability of winning is 4/36 The probability of losing is ____ The odds of winning are 4/36:32/36 = 4:32 = ___ : ___ 29 Example: Odds The odds of winning a game are 5:31. What is the probability of winning and losing? Suppose that you played 5+31 = 36 times. You would expected to win 5 times and lose 31 times. The probability of winning is 5/36. The probability of losing is 31/36. 30 Suppose the odds of winning are given as a:b. We want to the probability of winning. Algebraically, suppose p:q = p:(1-p) = a:b Treat ":" like it was a "/“ p/(1-p) = a/b bp = a(1-p) bp = a - ap ap + bp = a (a+b)p = a p = a/(a+b) 31 Example: Odds The odds of winning first prize in a raffle are 1:1999. What is the probability of winning? Suppose that 1+1999 = 2000 tickets are sold. We would expect to win 1 time out of 2000 The probability of winning is 1/2000. Using the formula: a = ___, b = _____ The probability is p = a/(a + b) = 1/(1 + 1999) = _________ 32 Example: Raffle The prize list for a raffle is Prize Number Value Odds New car (Kia) 1 $10,000 1:9999 TV set 10 $300 1:999 Meal for two 20 $50 1:499 Determine: – Expected value of a ticket – Number of tickets sold – If all of tickets costing $2.50 each are sold and if there is an addition cost of $2000 for printing and advertising, write a budget for the sponsors – Solution: See http://www.cs.plu.edu/courses/csce115/spring05/download/Pro bability/Raffle.xls 33

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