Not So Great Expectations! Which game do you think has the highest return per dollar?

For every dollar spent, you would expect to LOSE… Blackjack Blackjack cents Craps Craps cents Crossword Crossword -50 cents Heart and Stroke Heart and Stroke - 65 cents

Expected Value E(X) Chapter 4 – Supplemental (also Chapter 5) Mathematics of Data Management (Nelson) MDM 4U

Expected Value A property of an object or game of chance The value to which the average tends after a large number of trials If all outcomes are the same, it is the average of all possible values

Expected Value Average of all sides is 3.5 E(X) = 3.5 Average of all sections is 2595 E(X) = 2 595

What if the outcomes are not equally likely? You must factor in the probability as well as the value. For a game with outcomes x 1, x 2, …, x n That have probabilities p 1, p 2, …, p n E(X) = p 1 x 1 + p 2 x 2 + … + p n x n In English…multiply the value of every outcome by its probability and add them all up.

Expected / Average Value The probability-weighted sum of the possible values The long-run average value of an experiment over many repetitions Does NOT mean the "most likely value“ Is often not a value that the random variable can take on  May be unlikely or even impossible  The Expected Value of One die is 3.5 (impossible) Two dice is 7 (possible) Three dice is 10.5 (not possible)

Example 1 – “Nevada” tickets 2700 tickets per box - sold for $0.50 each 276 prizes as follows:  n($100) = 4  n($50) = 2  n($25) = 6  n($5) = 4  n($1) = 260 What is the probability of buying a winning ticket? P(win) = 276 / = 10.22% Would you be equally happy to win $1 as a $100?  How do these factor into the winning scheme?

Example 1 – Probabilities First, find the probability of each outcome. n($100) = 4P($100) = 4/2700 n($50) = 2P($50) = 2/2700 n($25) = 6P(25) = 6/2700 n($5) = 4P($5) = 4/2700 n($1) = 260P($1) = 260/2700

Example 1 – Expected Value To find the Expected Value, multiply every putcome by its probability and add them together: = = or $0.34 The Expected Value of a ticket is $0.34. However, the ticket costs $0.50. So you would expect to lose $0.16 on every ticket you buy.

Example 2 – Rolling a Die Suppose you play a game with a friend where you roll a fair die. If the roll is odd, your friend pays you the amount shown on the die. If the roll is even, you pay your friend the amount shown on the die. How much money would you expect to win or lose on each roll? Is this a fair game?

Example 2 – Rolling a die (cont’d) Odd numbers represent wins and have a positive value Even numbers represent losses and have a negative value E(X) = (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 = -3/6 = -1/2 or -$0.50 So you would expect to lose $0.50 on every play.

So looking at Expected Value, the games look like this: Blackjack Blackjack E(X) = Craps Craps E(X) = Crossword Crossword E(X) = 0.5 Heart and Stroke Heart and Stroke E(X) = 0.35