Question 1161243
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You clearly know HOW to calculate an expected value.  But in the work you show you are oversimplifying the problem.<br>
A couple of comments before I start discussing the approach to solving the problem: don't convert each part of the answer to decimal form.  Leave each part of the answer in the form n/52; add all those fractions, and wait until you have that total fraction before converting to decimal.<br>
Better yet, since every one of those fractions has denominator 52, ignore the denominator until the very end and divide by the 52 once to get your final decimal expected value.<br>
For example, for the aces, the calculation (ignoring the "52" denominator) is not simply (1)(4), because the ace of clubs is worth double and the ace of spades is worth triple.  So the calculation is 1(1+1+2+3) = 7(1)) = 7.<br>
Then you need to consider the suits when doing the calculations for the other cards.<br>
For the face cards, there are 3 each in hearts and diamonds, 3 in clubs, and 3 in spades; the calculation is 20(3+3+3(2)+3(3)) = 7(20+20+20) = 7(60).<br>
Then you will have similar calculations for the 2 through 9 cards:
2's: 2(1+1+2+3) = 7(2)
3's: ... = 7(3)
...
9's: ... = 7(9)
10's: ... = 7(10)<br>
Add all those pieces together and divide by 52; that is the expected value of your winnings.<br>
Then of course if the game is EXACTLY "fair", that expected value should be the cost of playing the game.  However, it is unlikely that the expected value will turn out to be a whole number of dollars, or even a whole number of cents -- so there probably won't be a price for playing the game that is EXACTLY fair.<br>