Probability Part #3. Expected Value

Transcription

1 Part #3 Expected Value

2 Expected Value expected value involves the likelihood of a gain or loss in a situation that involves chance it is generally used to determine the likelihood of financial gains and losses in situations such as lotteries, business, and multiple choice tests. an example of expected value would be when you spend money in a slot machine or VLT, if you put 50 cents into the machine 100 times, how much money would the government expect you to receive?

3 Calculating Expected Value Step 1 - determine the probability of the required events Step 2 - find the gain or loss associated with each event (the payoff of each event) Step 3 - multiply the probability of each event by its payoff Step 4 - find the sum of the probability of each event multiplied by its payoff, this is the expected value

4 Consider the following game, you have a 1 in 5 chance of winning the game and a 4 in 5 chance of losing. The game costs $1 each time you play. If you win, you receive $4. If you lose the game, you receive nothing. Find the expected value of the game. Event Amount Won Cost of Play Pay Off x Payoff You may find it helpful to complete a chart

5 Event Amount Won Cost of Play Pay Off x Payoff Expected Value =

6 What Does it Mean? The expected value means that if you played this game many times, on average, you would lose 20 cents for each time you played If you played this game 10 times you can expect to lose $2 If you played this game 20 times you can expect to lose $4 If you played this game 100 times you can expect to lose $20 However, if you play the game only once, you would either gain $3 or lose $1

7 What Does it Mean? In general, the following is true about expected value in games: If you play any game with an expected value less than 0, you can expect to lose money If you play any game with an expected value of 0 you can expect to break even, this is called a fair game If you play any game with an expected value of greater than 0, you can expect to gain money Games on VLTs and games such as lotteries always have an expected value that is negative so they can make money for the organization running the game (at the player s expense)

8 Consider the following game. It costs $3 each time you play. You have a 1 in 10 chance of winning $25, a 1 in 5 chance of winning $5 and a 7 in 10 chance of receiving nothing. find the expected value if you play this game many times, will you gain or lose money is this a fair game? Explain

9 Event Amount Won Cost of Play Pay Off x Payoff Expected Value =

10 In order to raise money for cancer research at the St. Boniface Hospital, the St. Boniface Hospital Foundation decides to have a charity lottery. The grand prize is $15,000. There are also 10 additional prizes of $500. Each ticket costs $15. Only 10,000 tickets will be sold. What is the expected value if you buy one ticket? Is this a fair game? Why or why not? How much will the St. Boniface Hospital Foundation gain or lose from this charity event? Why would the number of tickets sold be limited to 10,000?

11 Event Amount Won Cost of Play Pay Off x Payoff Expected Value =

12 Nicholas decides that he doesn t need to go to school anymore, Instead, he is going to win the lottery by choosing 7 random numbers and a bonus number. The draws for this lottery are on Wednesday and Friday. Therefore, Nicholas wants to bid two tickets a week to increase his chance os winning. Nicholas buys two $5 tickets every week for a year. The probabilities of winning a prize based on a $5 ticket are shown on the next slide. On average, how much will Nicholas have won or lost after the year is over?

13 Matches Needed to Win Prize 7 of 7 $20,000,000 1:28,633,528 6 of 7 + bonus $50,000 1:4,090,504 6 of 7 $25,000 1:99, of 7 $500 1:1, of 7 $20 1: of 7 + bonus $20 1: of 7 $5 1:8.1 2,1 of 7 $

14 Expected Value in Business Expected value is not only used for gaming but can also be beneficial for making business decisions You can continue to use expected value charts to determine the answers but it is not mandatory to do so

15 Based on past experience, a building contractor sets her probability of winning a contract at The contract is worth $25,000, and she determines it will cost her $2,400 to prepare a contract proposal Find the expected value of the contract proposal Is it financially a good ideas for her to bid on the contract? What other factors might she consider before deciding whether to bid on the contract?

16 Event Amount Won Cost of Play Pay Off x Payoff Expected Value =

17 What Does it Mean? The expected value does not mean that if the contractor bids on the job she will receive $5,100. It means that if she bids on many such contracts, the expected gain per contract is $5,100 For example, if she bids on 10 such contracts, on average her gain will be 10($5,100) = $51,000. If she bids on only one contact, she will either gain $22,600 or lose $2,400, depending on the results.

18 Consider the following multiple choice test. Each question on the test has five possible answers. You receive 3 points for each correct answer but lose 1 point for each incorrect answer. What is you expect value on a question you answer by randomly guessing? Is it to your advantage to answer a question by randomly guessing? What is your expected value on a question in which the correct response is one of two potential answers? Is it to your advantage to answer a question in which the correct response is only of two potential answers?

19 Event Amount Won Cost of Play Pay Off x Payoff Expected Value =