1 Mathematical Expectation Mathematical Expectation Ernesto Diaz, Mathematics Department Redwood High School

Mathematical Expectation What is it? An average. Example: You buy tires rated for 60,000 miles. This is the expected life of the tires. Of course, your experience may differ depending on driving habits and random chance.

Mathematical Expectation What is it used for? Typically, as a tool for decision making. Definition: E = a 1 p 1 + a 2 p 2 + … + a n p n The a i ’s are values of individual outcomes. The p i ’s are probabilities of those outcomes.

An (contrived) Example Given: Bet on a 4-horse race, with pay-off $100 if Horse 1 wins and $50 if horse 2 wins. Nothing if horses 3 or 4 win. What is the expected value of your bet? Note: a 1 = $100, a 2 = $50, a 3 =a 4 =$0 p 1 = p 2 = p 3 = p 4 = 1/4 (assumed) Answer: E = $100(1/4) + $50(1/4) + $0(1/4) + $0(1/4) = $25 + $12.50 = $37.50an unusually good bet!

Expected Value The symbol P 1 represents the probability that the first event will occur, and A 1 represents the net amount won or lost if the first event occurs.

Example Teresa is taking a multiple-choice test in which there are four possible answers for each question. The instructor indicated that she will be awarded 3 points for each correct answer and she will lose 1 point for each incorrect answer and no points will be awarded or subtracted for answers left blank. If Teresa does not know the correct answer to a question, is it to her advantage or disadvantage to guess? If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at the answer?

Solution Expected value if Teresa guesses.

Solution continued—eliminate a choice

Example: Winning a Prize When Calvin Winters attends a tree farm event, he is given a free ticket for the $75 door prize. A total of 150 tickets will be given out. Determine his expectation of winning the door prize.

Example When Calvin Winters attends a tree farm event, he is given the opportunity to purchase a ticket for the $75 door prize. The cost of the ticket is $3, and 150 tickets will be sold. Determine Calvin’s expectation if he purchases one ticket.

Solution Calvin’s expectation is  $2.49 when he purchases one ticket.

Fair Price Fair price = expected value + cost to play

Example Suppose you are playing a game in which you spin the pointer shown in the figure, and you are awarded the amount shown under the pointer. If is costs $10 to play the game, determine a) the expectation of the person who plays the game. b) the fair price to play the game. $1 0 $2 $2 0 $1 5

Solution $0 3/8 $10 $5  $8 Amt. Won/Lost 1/8 3/8Probability $20$15$2 Amt. Shown on Wheel

Solution Fair price = expectation + cost to play =  $ $10 = $8.87 Thus, the fair price is about $8.87.

16 Why the House Wins GameExpected value for $1 bet Baccarat Blackjack Craps Slot machines Keno (eight-spot ticket) Average state lottery -$ to + $0.10 (varies with strategies) -$0.014 for pass, don’t pass, come, don’t come bets only -$0.13 to ? (varies) -$0.29 -$0.48 There isn’t a single bet in any game of chance with which you can expect to break even, let alone make a profit