Expected Value Examples

At the carnival, we found a bean bag toss booth. Giant stuffed monkey, here we come. There's a 70% chance of throwing the bean bag into the outer ring, a 20% chance of throwing it into the middle ring, and a 10% chance of throwing it into the center of the target. We get 1 ticket for the outer ring, 2 tickets for the middle ring, and 3 tickets for the center. What is the expected value (number of tickets) for this game?

The expected value tells us the long-term average result for some event. This time, we're winning tickets from a carnival game.

EV = x₁p₁ + x₂p₂ + x₃p₃

There are 3 possible outcomes: throwing the bean bag into the outer, middle, or center rings. We know the probability, p_i, of each outcome happening, and we know how many tickets, x_i, we would get in each case. We just multiply those two things together, then add them up for all the possible outcomes.

EV = 1(0.70) + 2(0.20) + 3(0.10)

= 0.7 + 0.4 + 0.3

= 1.4 tickets

We expect to win 1.4 tickets per game played, on average. That's only 715 games to get our hands on that monkey.

The local Kiwanis club is hosting 350 raffle ticket sale where the grand prize winner gets a $7000 Polaris 4-wheeler. Each ticket costs $50. If you're a cold-hearted miser, unconcerned with giving to a good cause, would you buy a ticket to try and get the prize?

The question is, is the expected value from buying a ticket greater than the $50 you initially spend? The possible outcomes are winning the prize, or not winning. Losing gets you nothing, while winning is worth $7000, plus the knowledge that you've deprived someone else of winning.

Remember, we're assuming you're completely cold-hearted here.

The probability of winning is 1 out of 350, because each ticket has an equal chance of being picked. We can now set up the expected value equation.

Our expected value is $20, but it cost $50 to buy a ticket. That is not a sound investment, so you would cruelly turn your back on a charitable cause, you monster.

We recently won the lottery, and we're looking to invest some of that primo cash. Miss Auntra Panure approached us with this deal: invest in her company and the probability for success is 20%. The payout, if successful, would be $20 million. All she's asking for is an up-front investment of 25% of the payout, or $5 million.

Mr. E. Z. Munee says he can do better than Miss Panure's deal. He's sure his new company has at least a 55% percent of succeeding, and we'll double our money if we just make a simple one-time investment of $1 million.

Who has the better offer? Is either one worth taking?

There's a lot of information given here, but at the end of the day we want to know if we can expect to make any money playing the odds. Let's assume that all the numbers given to us are correct. Then we just need to know the expected value for each investment.

For Miss Auntra Panure, the outcome is getting $20,000,000. Cha-ching. However, the probability for that outcome is 0.2. Multiplying those together gets us:

EV₁ = 0.2(20,000,000) = $4,000,000

The expected value is less than our $5 million initial investment. Even though it would be possible to make money here, it isn't a safe investment.

Now for Mr. E. Z. Munee's offer. His payout is $2,000,000. Cha-ching again. The probability of success, 55%, is much better. The expected value here is:

EV₂ = 0.55(2,000,000) = $1,100,000

Now our expected value is more than our investment, by $100,000. That's nothing to sneeze at. In the end, Miss Panure's deal is a pile of manure, while Mr. E. Z. Munee's is easy money. Yes, their names were puns. No, we feel no shame over it.