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.

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.

Possible Losses/Gains

Probability P(losses/gains)

-$3

-$2

$12

6

3

6

2

6

1

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: _______________

50.1$6

9

6

33$

67.0$6

4

6

22$

00.2$6

12

6

112$

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.

17.0$00.2$67.0$50.1$

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.

Possible Outcome

Prize Amount

Possible Losses/Gains

Probability P(losses/gains)

Lose 0 -$1 97/100

3rd $1 $0 1/100

2nd $10 $9 1/100

1st $50 $49 1/100

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.

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.

$200

$600

$100 $400

$800$900

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$500The expected value of this spinner is $500.

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.

-$200

-$600

-$100 $400

$800

$300

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$100The 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

-$600 -$200 -$100 $300 $400 $800

-$600 -$1200 -$800 -$700 -$300 -$200 $200

-$200 -$800 -$400 -$300 $100 $200 $600

-$100 -$700 -$300 $0 $200 $300 $700

$300 -$300 $100 $200 $600 $700 $1100

$400 -$200 $200 $300 $700 $800 $1200

$800 $200 $600 $700 $1100 $1200 $1600

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 spinsTotal possible spins = 36So probability is 13/36