Expected Value in Games

Question

You and some of your friends decide to visit the Eastern Idaho state fair. One of your friends comes to you and says that he wants you to play this new game he found. He says he just played it and won $10 after playing 3 times.

Here are the game rules:
You just pay $3.00 to start the game and spin a big spinner. Whichever number it lands on, you get that much money back. The spinner has four sections labeled 1, 2, 3 and 4 each is a different size. See the picture below of what the spinner looks like.

You just finished taking FDMAT108 and remember learning about the expected value. You quickly calculate the expected value before you decide whether you will play or not. You find the probabilities of landing on each number and record them in a nice little table seen below.

Spinners Rules

     Spinning Probabilities
Land on Number       Probability
1                               0.377
2                                0.258
3                                0.203
4                               0.162


What would your expected earnings on average be per game?

About how much should you expect to win (or lose) if you played 100 times?

Answer 1

Step 1. Find the expected value of the amount you win on a single spin.

OutcomeProbabilityAmount won ($)Contribution to expectation
1	0.377	1	1×0.377=0.3771 × 0.377 = 0.3771×0.377=0.377
2	0.258	2	2×0.258=0.5162 × 0.258 = 0.5162×0.258=0.516
3	0.203	3	3×0.203=0.6093 × 0.203 = 0.6093×0.203=0.609
4	0.162	4	4×0.162=0.6484 × 0.162 = 0.6484×0.162=0.648

Now add these contributions:

0.377 + 0.516 + 0.609 + 0.648 = 2.150

Expected winnings from the spinner: $2.15

Step 2. Subtract the cost to play.

You pay $3.00 each time, so your expected net earnings are:

2.15 − 3.00 = − 0.85

Expected earnings per game = −$0.85

That means on average you lose 85 cents per play.

Step 3. Expected result after 100 games.

100 × (−0.85) = −85

Expected total after 100 games = −$85

Final Answer:

• Expected value per game: −$0.85

• Expected total after 100 games: Lose about $85

So, even though your friend got lucky, the math says this game is not in your favor.