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Expected Value

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Expected Value and Payoffs

When playing a game of chance there are three basic elements. There is the cost to play the game (usually), the probability of winning the game, and the amount you receive if you win. If games of chance with these three elements are played repeatedly, you can use probability and averages to calculate how much you can expect to win or lose in the long run.

Consider a dice game that pays you triple your bet if you roll a six and double your bet if you roll a five. If you roll anything else you lose your bet. What is your expected return on a one dollar wager?

Watch This

http://www.youtube.com/watch?v=j__Kredt7vY Khan Academy: Expected Value

Guidance

A weighted average is like a regular average except the data is often given to you in summary form.

Data in Raw Form:

1, 3, 5, 3, 2, 1, 2, 5, 6, 4, 5, 2, 6, 1, 4, 3, 6, 1, 2, 4, 6, 1, 3, 1, 3, 5, 6

Data in Summary Form:

Number Occurrence Count
1 6
2 4
3 5
4 3
5 4
6 5
Total Occurrences: 27

Notice that the summary data indicates, for example, how many times a 1 was rolled (6 times). To calculate the total number of occurrences of data:

  • In raw form: count how many data points you have
  • In summary form: find the sum the occurrence column

To calculate the average:

  • In raw form: find the sum of the data points and divide by the total number of occurrences.
  • In summary form: find the sum of the data points by finding the sum of the product of each number and its occurrence:

1 \cdot 6+2 \cdot 4+3\cdot 5+4 \cdot 3+5 \cdot 4+6 \cdot 5=91

Then, divide that sum by the total number of occurrences. In a sense, you are assigning a weight to each of the six numbers based on their frequency in your 27 trials. 

The same logic of finding the average of data given in summary form applies when doing theoretical expected value for a game or a weighted average. Consider a game of chance with 4 prizes ($1, $2, $3, and $4) where each outcome has a specific probability of happening, shown in the table below: 

Number Probability
$1 50%
$2 20%
$3 20%
$4 10%

Note that the probabilities must add up to 100%! In order to calculate the expected value of this game, weight the outcomes by their assigned probabilities.

\$1 \cdot 0.50+\$2 \cdot 0.20+\$3 \cdot 0.30+\$4 \cdot 0.10=\$2.20

This means that if you were to play this game many times, your average amount of winnings should be $2.20. Note that there will be no game that you actually get $2.20, because that was none of the options. Expected value is a measure of what you should expect to get per game in the long run.

The payoff of a game is the expected value of the game minus the cost. If you expect to win about $2.20 on average if you play a game repeatedly and it costs only $2 to play, then the expected payoff is $0.20 per game.

In general, to find the expected value for a game or other scenario, find the sum of all possible outcomes, each multiplied by the probability of its occurrence.

Example A

A teacher has five categories of grades that each make up a specific percentage of the final grade. Calculate Owen’s grade.

Category Weight Owen’s grade
Quizzes and Tests 30% 78%
Homework 25% 100%
Final 20% 74%
Projects 20% 90%
Participation 5% 100%

Solution: Using the concept of weighted average, weight each of Owen’s grades by the weight of the category.

  0.78 \cdot 0.3+1 \cdot 0.25+0.74 \cdot 0.20+0.90 \cdot 0.20+1 \cdot  0.05=0.862

Owen gets an 86.2%.

Example B

Courtney plays a game where she flips a coin. If the coin comes up heads she wins $2. If the coin comes up tails she loses $3. What is Courtney’s expected payoff each game?

Solution: The probability of getting heads is 50% and the probability of getting tails is 50%. Using the concept of weighted averages, you should weight winning 2 dollars and losing 3 dollars by 50% each. In this case there is no initial cost to the game.

2 \cdot 0.50-3 \cdot 0.50=-0.50

This means that while sometimes she might win and sometimes she might lose, on average she is expected to lose about 50 cents per game.

Example C

Paul is deciding whether or not to pay the parking meter when he is going to the movies. He knows that a parking ticket costs $30 and he estimates that there is a 40% chance that the traffic police spot his car and write him a ticket. If he chooses to pay the meter it will cost 4 dollars and he will have a 0% chance of getting a ticket. 

Is it cheaper to pay the meter or risk the fine?

Solution: Since there are two possible scenarios, calculate the expected cost in each case.

Paying \ the \ meter: \$ 4 \cdot 100 \% = \$ 4

Risking \ the \ fine: \$ 0 \cdot 60 \% + \$ 30 \cdot 40\%=\$ 12

Risking the fine has an expected cost three times that of paying the meter.

Concept Problem Revisited

In a game that pays you triple your bet if you roll a six and double your bet if you roll a five, the expected return on a one dollar wager is:

\$ 0 \cdot \frac{2}{3}+\$ 2\cdot \frac{1}{6}+\$ 3\cdot \frac{1}{6}=\frac{5}{6}

If you spend $1 to play the game and you play the game multiple times, you can expect a return of \frac{5}{6} of one dollar or about 83 cents on average.

Vocabulary

A weighted average is an average that multiplies each component by a factor representing its frequency or probability. 

The expected value is the return or cost you can expect on average, given many trials.

The payoff of a game is the expected value of the game minus the cost.

Guided Practice

1. What is the payoff of a slot machine that costs a dollar to play and pays out $5 with probability 4%, $10 with probability of 2%, and $30 with probability 0.5%?

2. What is the expected value of an experiment with the following outcomes and corresponding probabilities?

Outcome 31 35 37 39 43 47 49
Probability 0.1 0.1 0.1 0.2 0.2 0.2 0.1

3. Every day you record about how long it takes to get to school.

Time Number of Days
5-7 minutes 1
7-8 minutes 4
8-9 minutes 7
9-10 minutes 9
10-12 minutes 2

How long does it take you to get to school on average?

Answers:

1. 0 \cdot 0.935+5 \cdot 0.04+10 \cdot 0.02+30 \cdot 0.005-1=-0.45.  You will lose $0.45 on average if you play the slot machine many times.

2. 31 \cdot 0.1+35  \cdot 0.1+37  \cdot 0.1+39  \cdot 0.2+43  \cdot 0.2+47  \cdot 0.2+49  \cdot 0.1=41

3. To answer this question, you could find the weighted average of the expected value. If you choose expected value, consider that the situation gives you frequency rather than probability. You can calculate the probability of each of the categories by dividing each frequency by the total number of days (23). Since the time occurs in intervals, it is reasonable to use the average time in each interval as representative of the category when calculating the expected value.

Time Number of Days Probability
5-7 minutes 1 \frac{1}{23}
7-8 minutes 4 \frac{4}{23}
8-9 minutes 7 \frac{7}{23}
9-10 minutes 9 \frac{9}{23}
10-12 minutes 2 \frac{2}{23}

6 \cdot \frac{1}{23}+7.5 \cdot \frac{4}{23}+8.5 \cdot \frac{7}{23}+9.5 \cdot \frac{9}{23}+11 \cdot \frac{2}{23}=\frac{203}{23}\approx8.8

On average, it takes you 8-9 minutes to get to school.

Practice

1. Explain how to calculate expected value.

2. True or false: If the expected value of a game is $0.50, then you can expect to win $0.50 each time you play.

3. True or false: The greater the number of games played, the closer the average winnings will be to the theoretical expected value.

4. A player rolls a standard pair of dice. If the sum of the numbers is a 6, the player wins $6. If the sum of the numbers is anything else, the player has to pay $1. What is the expected value for this game?

5. What is the payoff of a slot machine that costs 25 cents to play and pays out $1 with probability 10%, $50 with probability of 1%, and $100 with probability 0.01%?

6. A slot machine pays out $1 with probability 5%, $100 with probability of 0.5%, and $1000 with probability 0.01%? If the casino wants to guarantee that they won’t lose money on this machine, how much should they charge people to play?

7. What is the expected value of an experiment with the following outcomes and corresponding probabilities?

Outcome 12 14 18 20 21 22 23
Probability 0.05 0.1 0.6 0.1 0.1 0.03 0.02

Calculate the final grades for each of the students given the information in the table.

Category Weight Sarah Jason Kimy Maria Kayla
Quizzes and Tests 30% 74% 85% 90% 80% 75%
Homework 25% 95% 40% 100% 90% 95%
Final 20% 68% 80% 85% 70% 50%
Projects 20% 85% 70% 95% 75% 85%
Participation 5% 95% 100% 100% 80% 60%

8. What is Sarah’s final grade?

9. What is Jason’s final grade?

10. What is Kimy’s final grade?

11. What is Maria’s final grade?

12. What is Kayla’s final grade?

13. Look back at the grades and final grades for the five students. Do the grades seem fair to you given how each student performed in each of the areas? Do you think the category weights should be changed?

14. You are in charge of a booth for a game at the fair. In the game, players pick a card at random from the deck. If the card is a J, Q, or K, the player wins $5. What is the minimum amount you should charge in order to feel confident you will make a profit by the end of the fair?

15. Make up your own game that has at least 2 possible outcomes with an expected payoff of $0.50.

16. Explain why it makes sense for a casino to consider the concept of expected value when designing their games.




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