SOLUTION: A certain game involves tossing 3 fair coins, and it pays 10 cents for 3 head, 5 cents for 2 heads, and 3 cents for 1 head. Is 5 cents a fair price to pay to play this game ? (That

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Question 739730: A certain game involves tossing 3 fair coins, and it pays 10 cents for 3 head, 5 cents for 2 heads, and 3 cents for 1 head. Is 5 cents a fair price to pay to play this game ? (That is, does the 5 cents cost to play make the game fair?)
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
A certain game involves tossing 3 fair coins, and it pays 10 cents for 3 head, 5 cents for 2 heads, and 3 cents for 1 head. Is 5 cents a fair price to pay to play this game ? (That is, does the 5 cents cost to play make the game fair?)
First we determine the various probabilities

TOSS   P(TOSS)   Number of heads

HHH     1/8           3
HHT     1/8           2
HTH     1/8           2
HTT     1/8           1
THH     1/8           2
THT     1/8           1
TTH     1/8           1
TTT     1/8           0

Then we make this chart:

Number of heads   Prob   Winning   Expectation
       x            p      
       3          1/8      10¢      (1/8)10¢ = 1.25¢
       2          3/8       5¢      (3/8)5¢ =  1.875¢
       1          3/8       3¢      (3/8)3¢ =  1.125¢ 
       0          1/8       0¢      (1/8)0¢ =  0¢ 
                                  ---------------------
                                 Total expectation = 4.25¢

So if you played this game many times you would average 
getting 4.25¢ per time you played.

So if you paid a nickel to play, you would average losing
0.75¢ per game, (three-fourths of a penny!) so it would not be fair.

Edwin
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