Question 1203216
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Answer: <font color=red size=4>$2.00</font>


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Explanation:


X = profit in dollars
P(X) = probability of getting a certain profit X


X takes on exactly two values
Either X = 769 happens when you get all four guesses correct. 
Or X = -1 to represent losing $1 when getting at least one guess wrong (i.e. one or more guesses wrong)


There are 4 suits. The probability of getting a suit correct on any guess is 1/4.
Getting 4 correct guesses in a row has probability (1/4)^4 = 1/256


The complement of this is:
1 - 1/256 = 256/256 - 1/256 = (256-1)/256 = 255/256
which represents the probability of losing.


To summarize so far:
P(X) = 1/256 when X = 769
P(X) = 255/256 when X = -1


<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td></tr><tr><td>-1</td><td>255/256</td></tr><tr><td>769</td><td>1/256</td></tr></table>


Form a new column labeled X*P(X)
As the label suggests, we'll multiply each X and P(X) value for any particular row.
<table border = "1" cellpadding = "5"><tr><td>X</td><td>P(X)</td><td>X*P(X)</td></tr><tr><td>-1</td><td>255/256</td><td>-255/256</td></tr><tr><td>769</td><td>1/256</td><td>769/256</td></tr></table>


E(X) = expected value often denoted as mu = expected profit in this case
E(X) = sum of X*P(X) values
E(X) = (-255/256) + (769/256)
E(X) = (-255+769)/256
E(X) = 514/256
E(X) = 2.0078125
E(X) = <font color=red>2.00</font> when rounding to the nearest dollar.


When rounding to the nearest cent, we get $2.01 as the average profit.
Because this profit value is positive, the player should keep playing because the odds are tilted in their favor. 


Realistically the odds are usually tilted in favor of the house (aka the person(s) who runs the game).
Otherwise, casino owners would be quickly out of business.


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Another approach:


Let's say the person plays 256 games.
Each game is defined as "guess the suit of 4 cards" (meaning 256 games involves 4*256 = 1024 cards).


When X = 769, it leads to P(X) = 1/256. 
The player would be expected to win $769 exactly one time out of those 256 attempts.
The other 255 attempts the player loses $1 each (totaling -255 in losses).
Wins or losses aren't guaranteed for any particular game. This is just a theoretical thought experiment.


This helps explain the previous calculation (-255+769)/256 mentioned earlier.


The player would net -255+769 = 514 dollars over those 256 attempts.
Their average profit would be 514/256 = 2.0078125 dollars which rounds to <font color=red>$2.00</font>
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