Question 1204551: Suppose it costs $57 to roll a pair of dice. You get paid 8 dollars times the sum of the numbers that appear on the dice. What is the expected payoff of the game? Is it a fair game?
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
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Here are the possible dice sums when considering two 6 sided dice:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
If you were to look at a dice sum frequency table, or histogram, notice how the most frequent value is right in the middle.
That would be the sum 7.
Furthermore, take note of the symmetry on either side of 7.
Another way to reach 7 is to find the average of the set {1,2,3,4,5,6} which is 3.5; this represents the average value of any single roll of one die.
The 3.5 doubles to 7 when dealing with two dice.
So the average dice roll is 7.
The expected payout is 8*7 = 56 dollars.
This is one dollar less than the price to play the game ($57), which means the game is not fair.
The player loses $1 on average per roll.
The winnings must equal the cost to play the game for the game to be fair.
In a fair game, the player neither gains money nor loses money.
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