Please check my answers.

Thank you!

Another card game. In a new card game, you start with a well-shuffed full deck and draw 3 cards without replacement. If you draw 3 hearts, you win $50. If you draw 3 black cards, you win $25. For any other draws, you win nothing.
(a) Create a probability model for the amount you win at this game, and nd the expected winnings. Also compute the standard deviation of this distribution.

My answer:
E(x)= 3.60
Standard deviation= 9.65

(b) If the game costs $5 to play, what would be the expected value and standard deviation of the net pro t (or loss)? (Hint: profit = winnings - cost; X - 5)

My answer:
3.60-5= -1.40

(c) If the game costs $5 to play, should you play this game? Explain.

My answer:
No.. because I would lose $1.40.

To check your answers for this card game problem, let's go through the steps to find the expected winnings and standard deviation as well as the net profit (or loss).

(a) Probability Model and Expected Winnings:
To create a probability model, we need to calculate the probability of each outcome and multiply it by the corresponding winnings.

The number of ways to draw 3 hearts from a full deck is given by the combination formula: C(13, 3) = 286. The probability of this outcome is 286/C(52, 3).

Next, the number of ways to draw 3 black cards is given by C(26, 3) = 2600. The probability of this outcome is 2600/C(52, 3).

For any other draw, there are C(39, 3) = 9,139 possible outcomes. The probability is 9139/C(52, 3).

Let's calculate the probabilities and expected winnings:

Probability of drawing 3 hearts = 286/C(52, 3)
Probability of drawing 3 black cards = 2600/C(52, 3)
Probability of any other draw = 9139/C(52, 3)

Expected winnings = (Probability of drawing 3 hearts * $50) + (Probability of drawing 3 black cards * $25) + (Probability of any other draw * $0)

Now, calculate the probabilities and expected winnings to verify if your answer is correct.

Standard Deviation:
To calculate the standard deviation, you need to find the variance first. The variance is the sum of the squared differences between each outcome and the expected value, multiplied by their corresponding probabilities.

Variance = [(outcome1 - expected value)^2 * probability1] + [(outcome2 - expected value)^2 * probability2] + ...

Square root of variance = Standard deviation

Calculate the variance and standard deviation to verify if your answer is correct.

(b) Net Profit (or Loss):
To find the net profit, subtract the cost of playing the game ($5) from the expected winnings.

Net profit = Expected winnings - Cost

Calculate the net profit to verify if your answer is correct.

(c) Decision:
If the net profit is positive, it means you would make a profit, and if it is negative, it means you would incur a loss. If the net profit is zero, it means you're breaking even.

Based on your calculations, if the net profit is negative, then your answer "No... because I would lose $1.40" seems correct. However, double-check your calculations to confirm.

Remember, the calculations need to be accurate for a definitive answer.