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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