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.
(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)
(c) If the game costs $5 to play, should you play this game? Explain.
To create a probability model for the amount you win at this game, we need to determine the probability of each possible outcome and assign the corresponding winnings.
(a) Probability Model:
Let's calculate the probabilities of different outcomes:
1. Drawing 3 hearts: There are 13 hearts in a deck of 52 cards, so the probability of drawing a heart on the first card is 13/52. After the first heart is drawn, there are 12 hearts left in a deck of 51 cards, so the probability of drawing a heart on the second card is 12/51. Similarly, the probability of drawing a heart on the third card is 11/50. Multiply these probabilities to get the probability of drawing 3 hearts: (13/52) * (12/51) * (11/50).
2. Drawing 3 black cards: There are 26 black cards in a deck of 52 cards, so the probability of drawing a black card on the first draw is 26/52. After the first black card is drawn, there are 25 black cards left in a deck of 51 cards, so the probability of drawing a black card on the second draw is 25/51. Similarly, the probability of drawing a black card on the third draw is 24/50. Multiply these probabilities to get the probability of drawing 3 black cards: (26/52) * (25/51) * (24/50).
3. For any other draws, the probability is simply 1 - (probability of drawing 3 hearts) - (probability of drawing 3 black cards).
Now, assign the corresponding winnings:
- Drawing 3 hearts: $50
- Drawing 3 black cards: $25
- Any other draws: $0
To find the expected winnings, multiply each outcome's probability with its corresponding winnings and sum them up:
Expected winnings = (probability of drawing 3 hearts) * $50 + (probability of drawing 3 black cards) * $25 + (probability of any other draws) * $0
To compute the standard deviation of this distribution, we need to calculate the individual variances and then take the square root of their sum.
(b) To calculate the expected value and standard deviation of the net profit (or loss) when the game costs $5 to play, subtract the cost from the expected winnings:
Net Profit = Expected winnings - Cost
(c) Whether you should play this game or not depends on the expected value of the net profit (or loss). If the expected net profit is positive, it indicates a favorable outcome on average and suggests that playing the game might be worth it. However, if the expected net profit is negative, it implies a likely loss on average, and it would be better to avoid playing the game.
By calculating the expected value and standard deviation, we can determine if playing the game is advisable or not.