# 3.2: Applied Expected Value Calculations

**At Grade**Created by: Bruce DeWItt

(Note - Three Separate Videos for this Section)

### Learning Objectives

- Understand the concept of a fair game
- Be able to analyze a game of chance by building a probability model and calculating expected values from scratch

Casinos have a very delicate balancing act they must manage. First of all, they want to make money. However, people just don't like to lose money. In order to make money, the casino games have to be in favor of the house and not the player. Why don't casinos tilt the games even more to the house's favor? If they did, their expected value would certainly go up. On the other hand, attendance at the casino would go down.

No matter what the odds, a casino can't make money unless they can keep people coming through the doors. Setting the games up so that there are still winners, some occasionally big, is good for attendance. You might even know someone who has made a large amount of money at a casino.

In this section, we will bring together our ideas about calculating probabilities from Ch. 2 along with the concept of expected value in order to be able to analyze a game of chance. We begin where we left off in section 3.1. A **fair game** is a game in which neither the player nor the house has an advantage. In other words, when all is said and done, the average player will not have made or lost any money whatsoever.

#### Example 1

A bag has 10 red marbles and 8 blue marbles in it. A player reaches into the bag pulling out 2 marbles, one after the other without replacement. If the color of the two marbles match, the player wins $10. If they don't match, the player wins nothing. The game costs $5 to play.

a) Use a tree diagram to help find the probability that the two marbles match. Use your result to build a probability model.

b) Is this game a fair game? If so, explain why. If not, give the value that the game should cost in order to be fair.

#### Solution

Begin with a tree diagram that shows what might happen when two marbles are pulled.

Summarizing these results, we get the probability model shown in Table 3.8.

Marble Distribution Result Red,Red Red,Blue Blue,Red Blue,Blue Value $10 $0 $0 $10 Probability \begin{align*}\frac{90}{306}\end{align*} \begin{align*}\frac{80}{306}\end{align*} \begin{align*}\frac{80}{306}\end{align*} \begin{align*}\frac{56}{306}\end{align*}

\begin{align*}EV=\left (\$10\right )\left (\frac{90}{306}\right )+\left (\$0\right )\left (\frac{80}{306}\right )+\left (\$0\right )\left (\frac{80}{306}\right )+\left (\$10\right )\left (\frac{56}{306}\right )=\frac{\$1,460}{306}\approx\$4.77\end{align*}

The expected value is $4.77. Notice, however, that $4.77 is the expected amount that the house

pays outeach game. The expected value for the house is $0.23 because every player must pay $5 to play. At $5, the game is not fair because it favors the house by an average of 23 cents every time the game is played. To be a fair game, it should cost $4.77.

The game of *GREED* is a game of chance in which players try to decide when they have accumulated enough points on a turn to stop. Two 6-sided dice are rolled. The player gets to keep the total that shows on the two dice. After every roll, the player can either decide to roll again and try to add to their current total for that turn or stop and put their points in the bank. The only catch in this game is that if a total of 5 is rolled, all points accumulated on that turn are lost. For example, suppose the first roll has a total of 9 and the player decides to go again. The next roll has a total of 7. The player now has 16 points accumulated on this turn and must decide to either put those 16 points in the bank or risk them. If they decide to risk the 16 points and a total of 5 comes up next, the score for that turn will be 0.

#### Example 2

Suppose a person is playing *GREED* and has accumulated 26 points so far. Is it to their advantage to roll one more time?

#### Solution

We will build a probability model and calculate the expected value based upon what might happen with one more roll. (See Example 3 from Section 3.1.) For example, there is a \begin{align*}\frac{1}{36}\end{align*} chance that the total will be 2. This would mean the player would have a total of 28 points with one more roll. The highest a player could have after this turn would be 38 points if they happen to roll a total of 12. The risk is that the player will roll a total of 5 and lose their 26 points.

The one item to be careful about here is that it is impossible to get a total of 31 in the chart. Remember, if you roll a total of 5, you lose all of your points. When we perform our expected value computation for the probability model above, we get approximately 29.6. In other words, if we roll exactly one more time, our average result will be almost 30 points. This is definitely better than stopping with 26 points. It is to the advantage of the player to roll again.

#### Example 3

An investor is going to make a long-term investment in a company. If all goes well, an investment of $100 will be worth $900 in twenty years. The risk is that the company may go bankrupt within twenty years in which case the investment is worthless. Suppose there is a 25% chance that the company will go bankrupt within 20 years. What is the expected value of this investment?

#### Solution

Start by building a probability model as shown below that shows that there is a 25% chance of making nothing and a 75% chance of making $900.

\begin{align*}EV=\left (\$0 \right )\left ({0.25} \right )+\left (\$900 \right )\left ({0.75} \right )=\$675\end{align*}. Taking into account that this investment cost $100, the investor should get an average profit of $575.

### Problem Set 3.2

(Note - Two Videos for the Following Problems)

#### Exercises

1) In the carnival game W*iffle Roll,* a player will roll a wiffle ball across some colored cups. Suppose that if the ball stops in a blue cup, the player wins $20. If it stops in a red cup, they win $10, and if it stops in a white cup, the player wins nothing. There are 25 white cups, 4 red cups, and 1 blue cup. Assume the chances of stopping in any cup is the same. How much should this game cost if it is to be a fair game?

2) In a simple game, you roll a single 6-sided die one time. The amount you are paid is the same as the amount rolled. For example, if you roll a one, you get paid $1. If you roll a two you get paid $2 and so on. The only exception to this is if you roll a 6 in which case you get paid $12. What should this game cost in order to be a fair game?

3) Suppose you are playing the game of *GREED* as described in Example 2. You have accumulated a total of 55 points on one turn so far. Is it to your advantage to roll one more time?

4) Suppose you are playing the game of *GREED* again. This time you have accumulated a total of 60 points in one turn so far. Is it to your advantage to roll one more time?

5) Using your results from numbers 4) and 5) and a little more investigation, for what number of points in a turn in the game of *GREED* does it make no difference if you roll one more time or stop? In other words, at what point total does the expected value with one more roll give the same total as if you had stopped?

6) In the Minnesota Daily 3 lottery, players are given a lottery ticket based upon 3 digits that they pick. If their 3 digits match the winning digits in the correct order, then the player wins $500. If the digits don't match, then the player loses. The game costs $1 to play. What is the expected value for a player of this lottery game.

7) A bucket contains 12 blue, 10 red, and 8 yellow marbles. For $5, a player is allowed to randomly pick two marbles out of the bucket without replacement. If the colors of the two marbles match each other, the player wins $12. Otherwise the player wins nothing. What is the expected gain or loss for the player?

8) An insurance company insures an antique stamp collection worth $20,000 for an annual premium of $300. The insurance company collects $300 every year but only pays out the $20,000 if the collection is lost, damaged or stolen. Suppose the insurance company assesses the chance of the stamp collection being lost, stolen, or destroyed at 0.002. What is the expected annual profit for the insurance company?

9) A prospector purchases a parcel of land for $50,000 hoping that it contains significant amounts of natural gas. Based upon other parcels of land in the same area, there is a 20% chance that the land will be highly productive, a 70% chance that it will be somewhat productive, and a 10% chance that it will be completely unproductive. If it is determined that the land will be highly productive, the prospector will be able to sell the land for $130,000. If it is determined that the land is moderately productive, the prospector will be able to sell the land for $90,000. However, if the land is determined to be completely unproductive, the prospector will not be able to sell the land. Based upon the idea of expected value, did the prospector make a good investment?

10) A woman who is 35 years old purchases a term life insurance policy for an annual premium of $360. Based upon US government statistics, the probability that the woman will survive the year is .999057. Find the expected profit for the insurance company for this particular policy if it pays $250,000 upon the woman's death.

11) A bucket contains 1 gold, 3 silver, and 16 red marbles. A player randomly pulls one marble out of this bag. If they pull a gold marble, they get to pick one bill at random out of a money bag containing a $100 bill, five $20 bills, and fourteen $5 bills. If they pull a silver marble out of the bag, they get to pick one bill at random out of a bag containing a $100 bill, two $20 bills, and seventeen $2 bills. If your marble is red, you automatically lose. The game costs $5 to play.

a) Build a tree diagram for this situation.

b) Build a probability model for this situation.

c) Calculate the expected gain or loss for the player.

12) A spinner has four colors on it, red, blue, green, and yellow. Half of the spinner is red and the remaining half of the spinner is split evenly among the three remaining colors. A player pays some money to spin one time. If the spinner stops on red, the player receives $2. If it stops on blue, the player receives $4. If it stops on either green or yellow, the player wins $5. What should this game cost in order to be a fair game?

13) A bag contains 1 gold, 3 silver, and 6 red marbles. A second bag contains a $20 bill, three $10 bills, and six $1 bills. A player pulls out one marble from the first bag. If it is gold, they get to pick two bills from the money bag (without replacement). If it is a silver marble, they get to pick one bill from the money bag, and if the marble is red, they lose. The game costs $3 to play. Should you play? Explain why or why not.

14) Suppose the Minnesota Daily 3 lottery adds a new prize. You still get $500 if you match all three digits in order, but you can also win $80 if you have the three correct digits but not in the right order. The game still costs $1. What is the expected value for a player of this lottery game?

#### Review Exercises

15) Consider the partially complete probability model given below.

a) What is the value of 'X'?

b) What is the expected value for this situation.

16) The student council is starting to prepare for prom and decides to name a committee of 6 members. Suppose that they decide the committee will have 2 juniors and 4 seniors on it. In how many ways can the committee be selected if there are 8 juniors and 8 seniors from which to select?

17) Three cards are dealt off the top of a well-shuffled standard deck of cards. What is the probability that all three cards will be the same color?":

18) A student does not have enough time to finish a multiple choice test so they must guess on the last two questions. List the sample space of the possible guesses for the last two questions if each question has only choices a, b, and c.

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