Question 1203216: Please show steps of how to solve the question.
You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards.
You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw.
You pay $1 to play.
If you guess the right suit every time, you get your money back and $769.
Determine your expected profit of playing the game over the long term?
Expected profit = $
(round your answer to the nearest dollar)
Found 2 solutions by Theo, math_tutor2020:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the probbility of guessing the right suit on any draw of the cards is 13/52 because each suit has 13 cards that are members of that suit.
13/52 = 1/4
the probability of guessing the right suit, 4 times out of 4 tries, is 1/4 * 1/4 * 1/4 * 1/4 = (1/4)^4 = 1/256 * 770 = 3.0078125
you'll bet 1 dollar each time and your expected return is an average of 3.0078125 each turn.
profit is revenue minus expense.
average profit is therefore 2.0078125 dollars.
this is how it works out as far as i can tell.
assume that the probabilities hold and that you bet 256,000 times.
you will have spent 256,000 dollars (one dollar each time).
you will win 769 dollars (256,000 / 256) = 1,000 times for a total of 769,000 dollars.
you will get back an additional 1000 dollars (one dollar for each of the 1,000 times that you won).
your expenditure is 256,000
you get back 1,000 + 769,000 = 770,000
your profit is 770,000 minus 256,000 = 514,000.
divide that by 256,000 times that you bet and your average profit is equal to 514000 / 256000 = 2.0078125 dollars each time.
if i did this correctly, that's a very good bet.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: $2.00
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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
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.
| X | P(X) | X*P(X) | | -1 | 255/256 | -255/256 | | 769 | 1/256 | 769/256 |
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) = 2.00 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 $2.00
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