14 Expected Value
learn by watching
The links below will launch the video lessons in YouTube
- Setting Up Probability Distribution Tables ( 7 minutes 44 seconds)
- Calculating Expected Value ( 6 minutes 28 seconds)
- Expected Value in the Washington Lottery ( 8 minutes 38 seconds)
Try Exercises #1-4.
learn by reading
In the last lesson, we looked at what to do when working with equally-likely possibilities. Many times, however, our possibilities are NOT equally likely. When we are working with situations where different numerical outcomes have different probabilities, it is often helpful to come up with an expected value to use for planning and predicting.
Calculating expected value
The first step in calculating an expected value is to create a probability distribution table. In this table, we want to describe the different outcomes, then identify their values along with their probabilities. It is important that you include ALL possibilities — that is, your probabilities need to add up to 100% for this calculation to have any meaning. The expected value, then becomes kind of a like a weighted average. The result is what value you could expect if you repeated the event many, many times.
Example 1: Consider a game where you roll a single dice. If you roll a 1 or 2, you lose $100. If you roll a 3, nothing happens. If you roll a , or 5 you win $5. If you roll a 6, you win $500. Set up a probability distribution table for this game, and then calculate its expected value.
To create a probability distribution table, we need to outline our possible events, their $ values, and the probability of that event occurring.
For this game, our distribution table would look something like this (notice that if we add all the probabilities together, we get 100%):
| Event Outcome | Value | Probability of this .Outcome |
| Roll a “1 or 2” | – $100 | 2/6 |
| Roll a “3” | $0 | 1/6 |
| Roll a “4”, or” 5″ | $5 | 2/6 |
| Roll a “6” | $500 | 1/6 |
To calculate the expected value, we are going to multiply each outcome’s value by its probability. Then add up all the possible outcomes together.
Expected Value = (-$100)*(1/6) + ($0)*(1/6) + ($5)*(3/6) + ($500)*(1/6)
I recommend using a calculator, such as Desmos, where you can enter the entire expression in one step. Doing that, I find that the expected value for this game is $51.67.
So what does that mean? If I were to play this game many, many times, I would expect to win an average of $51.67. The game is rigged for the player to win. It is generally considered a good idea to play if/when an expected value is positive. [NOTE- if you are only playing the game once, anything goes! Even though the game is rigged for the players to win in the long run, it is reasonable to expect to lose $100. So make sure you can afford the consequences. But if you continue to play many, many times, you will come out ahead.]
insurance applications of expected value
An important application of expected values is in calculating insurance risk.
Example 2: Suppose a company wants to provide life insurance policies that provide $1,000,000 in the event of a death. The mortality rate (chance of death) for someone aged 55-59 in the US is 4%.
- Find the expected payout that the insurance company should plan for when selling this policy to a 55-59 year old person.
- If the insurance company charges $300 per month for this policy, will they cover their payout expenses? What is the expected profit/loss for the company?
- First, we want to create an expected value table for this situation. Notice that based on statistics, 0.8% of people in this category will die during a given year, meaning that that other 99.2% of people will not die. If a person dies, the company will need to provide $1,000,000 to their beneficiary. If the person does not die, the company does not pay out anything.
Event Outcome Payout Probability Person (aged 55-59) dies $1,000,000 0.8% Person (aged 55-59) does not die $0 99.2% Expected value is calculated by multiplying each outcome value by its probability, and then adding those results together. As always, be sure percentage probabilities are expressed in decimal form when doing calculations.
Expected value = ($1,000,000)(0.008) + ($0)(0.992) = $8000.
- The company should plan an output of $8000 per year per customer when they sell a policy to a 55-59 year-old person. This would amount to $8000 per year, or 8000/12 = $666.67 dollars per month. If the company only sells the policy for $300 per month, they should expect to lose an average of $366.67 per month. Not a great business plan. The company should either plan to significantly increase its policy price, or perhaps lower the payout that the policy will provide.
lottery example of expected value
If you play the Washington lottery, let’s take a look at the unequal outcomes and see how the expected value can shed some light on your chances to win.
Example 3: To play the Washington State lottery, you want to pick 6 numbers between 1-49. Depending on how many of your numbers match the day’s drawing, you will win different amounts of money:
-
- All 6 numbers match — you get the jackpot prize. Chances of winning are 1 in 6.99 million.
- Only 5 numbers match — you get $1000. Chances of winning are 1 in 27.1 thousand.
- Only 4 numbers match — you get $30. Chances of winning are 1 in 516.
- Only 3 numbers match — you get $3. Chances of winning are 1 in 28.3
- Create a probability distribution table for the lottery. Assume the today’s jackpot is $1.2 million.
- Find the expected winnings for playing the lottery.
- If it costs $.50 to buy a lottery ticket, what is a player’s actual expected win/loss?
- Keep in mind that the “chance of winning” is just a way to report the probability. Write your chances in fraction form when filling in your table. That will make it easier to calculate the expected value.
Event Outcome Value of Prize Probability Match all 6 numbers $1,200,000 1/6,990,000 Match 5 numbers $1000 1/27,100 Match 4 numbers $30 1/516 Match 3 numbers $3 1/28.3 Match 2, 1, or 0 numbers $0 (everyone else) (Although we can calculate the percentage for those tickets that do not win a prize, it really is not necessary since we will be multiplying that number by 0)
- To calculate the expected winnings: ($1,200,000)(1/6,990,000) + ($1000)(1/27,100) + ($30)(1/516) + ($3)(1/28.3) + ($0) = $0.37What does this mean? The lottery needs to plan to pay out $0.37 for every person that plays the lottery. Of course, that money is very unevenly distributed!
- Since a player had to first pay to buy the ticket, the player’s expected value is actually: $0.37 – $0.50 = -$0.13. That is, a player would be expecting to lose $0.13 cents every time they play (assuming they play many, many times). In a single play, it is possible (though unlikely) to win big. However, if you play many, many times, you will end up having lost money in the long run because the expected value is negative.
Unsurprisingly, the lottery is rigged for players to lose.
Exercises: Try These!
For each of the following problems, create a probability distribution table. Then use your table to answer the questions that follow.
Case Study 1:
Case Study 2:
Case Study 3:
Case Study 4:
