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MM1D2d: Use expected value to predict outcomes. Expected Value. The expected value is often referred to as the “long-term” average or mean . This means that over the long term of doing an experiment over and over, you would expect this average.

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## MM1D2d: Use expected value to predict outcomes

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**Expected Value**• The expected value is often referred to as the “long-term” average or mean . • This means that over the long term of doing an experiment over and over, you would expect this average. • To find the expected value or long term average, simply multiply each value of the random variable by its probability and add the products.**Example:**• Suppose that the following game is played. A man rolls a die. If he rolls a 1, 3, or 5, he loses $3, if he rolls a 4 or 6, he loses $2, and if he rolls a 2, he wins $12. What gains or losses should he expect on average? (What is his expected value?)**Step 1:**• We must find the probabilities of each outcome. We can make a chart to help us see this.**Step 2:**• Now, we multiply the probability for each outcome by the amount of money either gained or lost for that outcome. • Expected value of rolling a 1, 3, or 5: ________________ • Expected value of rolling a 4 or 6: ________________ • Expected value of rolling a 2: _______________**Step 3:**• To find the expected value for the entire game (the answer), simply add up the expected value for each outcome. • Expected Value = • This means that the man playing this game is expected to lose an average of $0.17 each game he plays.**Example:**• Suppose that there is a raffle. Each ticket of the raffle cost $1.00. There are 100 tickets sold for the raffle. The top prize is $50.00; second prize receives $10.00; and third prize receives $1.00. What gains or losses should you expect on average? (What is the expected value?)**Step 1:**• We must find the probabilities of each outcome. We can make a chart to help us see this.**Step 2:**• Now, we multiply the probability for each outcome by the amount of money either gained or lost for that outcome. • Expected value of losing: -$1(97/100)= -$97/100= -$0.97 • Expected value of 3rd: $0 (1/100) = $0 • Expected value of 2nd: $10 (1/100) = $10/100 = $0.10 • Expected value of 1st: $50 (1/100) = $50/100 = $0.50**Step 3:**• To find the expected value for the raffle(the answer), simply add up the expected value for each outcome. • Expected Value = -$0.97 +$0.00 + $0.10 + $0.50 = -$0.37 • This means that if you play this raffle you are expected to lose an average of $0.37.**$200**$400 $100 $900 $600 $800 Spinner Example • What is the expected value of this spinner? • To find the expected value, add all the amount together. Then divide by the number of slices.**Step 1:**• Add all the amounts on the spinner. • $100+$200+$400+$600+$800+$900 • $3000 • Now divide by 6 slices; because there is a 1/6 probability of landing on any particular piece. • $3000/6 • $500 • The expected value of this spinner is $500.**-$200**$400 -$100 $300 -$600 $800 Spinner Example • What is the expected value of this spinner? • To find the expected value, add all the amount together. Then divide by the number of slices.**Step 1:**• Add all the amounts on the spinner together • -$100-$200+$400-$600+$800+$300 • $600 • Now divide by 6 slices; because there is a 1/6 probability of landing on any particular piece. • $600/6 • $100 • The expected value of this spinner is $100.**Extension**• If you spin this same spinner 10 times, at the end what would you expect your outcome to be? How much money would you have? • $100 x 10 = $1000 • What is the probability of making at most $400 on a single spin? • There are 5 pieces of the circle that are $400 or less. • So probability is 5/6. • What is the probability of making at least $300 on a single spin? • There are 3 pieces of the circle that are $300 or more. • So probability is 3/6.**Extension**• If you spin the spinner once and receive $300 dollars, what is the probability of a sum of $600 after the second spin? • $600-$300=$300 • There is only one $300 spot, so the probability is 1/6. • What is the probability of spinning the spinner twice and having a sum of at least $400? • First need to make a chart of all the possible sums you can get from spinning the spinner twice.**Extension**• What is the probability of spinning the spinner twice and having a sum of at least $400? • No we have our table, count the number of spins that are $400 or more. • 13 spins • Total possible spins = 36 • So probability is 13/36

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