Expected Value in Games

Question

You and some of your friends decide to visit the Eastern Iowa 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 $2.50 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 MATH 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.

 

           SPINNING PROBABILITIES

Land on Number         Number Probability
1                                         0.374
2                                         0.250
3                                         0.200
4                                         0.176

 

 

1.What would your expected earnings on average be per game?

    ?(Round to the nearest cent.)

2. Should you expect to get the "expected value" in the first game?

Yes
No

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

    ?(Round to the nearest dollar.)

 

 

Answer 1

The expected value E(x) is the sum over all possible payouts of the amount paid times the probability of winning that amount.  

 

So E(x) = 0.374*1 + 0.250*2 + 0.200*3 + 0.176*4

 

= 0.374 + 0.5 + 0.6 + 0.704 = 2.178

 

You would not expect to get the expected value in the first game.  The expected value represents a long term average.  In a game of chance any single outcome is unpredictable.  

 

If you played 100 times you will spend 100*2.50 = $250 and you will expect to win 100*2.178 = $217.8 so your expected loss is 250 - 217.8 = $32.20 = $32 to the nearest dollar