Sarah draws cards from a deck of 52 cards. She receives $.40 for each heart, $.50 for an ace, and $.90 for an ace of hearts. If each draw costs $.15 should she play the game?
expected value=probability*reward
EV=.40*12/52 + .50*4/52 + .90(1/52)
= 1/52 * (4.80+2+.9)=7.60/52=.13
It is a losing game.
To determine whether Sarah should play the game, we need to calculate her expected payout.
First, we need to find the probabilities of drawing different cards:
- There are 13 hearts out of 52 cards, so the probability of drawing a heart is 13/52 = 1/4.
- There are 4 aces out of 52 cards, so the probability of drawing an ace is 4/52 = 1/13.
- Since there is only one ace of hearts, the probability of drawing an ace of hearts is 1/52.
Next, let's calculate the expected payouts for each scenario:
- For each heart, Sarah receives $0.40. So the expected payout for drawing a heart is (1/4) * $0.40 = $0.10.
- For each ace, Sarah receives $0.50. So the expected payout for drawing an ace is (1/13) * $0.50 ≈ $0.04.
- For each ace of hearts, Sarah receives $0.90. So the expected payout for drawing an ace of hearts is (1/52) * $0.90 ≈ $0.02.
Now, let's calculate the total expected payout by adding up the expected payouts for each scenario:
Expected payout = ($0.10 for hearts) + ($0.04 for aces) + ($0.02 for ace of hearts) = $0.10 + $0.04 + $0.02 = $0.16.
Since Sarah pays $0.15 per draw, her expected payout of $0.16 is greater than the cost per draw, which is $0.15. Therefore, she should play the game as she is expected to make a profit in the long run.
To determine whether Sarah should play the game, we need to calculate her expected earnings per draw and compare it to the cost per draw.
First, let's calculate the number of hearts in a deck of 52 cards. There are 13 hearts in a deck (since there are 13 cards of each suit).
Next, let's calculate the probability of drawing a heart. Since Sarah is drawing cards randomly from a standard deck, there is an equal chance of drawing any card, so the probability of drawing a heart is 13/52, which simplifies to 1/4.
Now, let's calculate the number of aces in a deck. There are 4 aces in a deck (one ace of each suit).
The probability of drawing an ace is 4/52, which simplifies to 1/13.
Finally, let's calculate the expected earnings per draw:
- For each heart, Sarah receives $0.40, so the expected earnings from hearts per draw is (1/4) * $0.40 = $0.10.
- For each ace, Sarah receives $0.50, so the expected earnings from aces per draw is (1/13) * $0.50 = $0.03846 (approx. $0.04).
- For the ace of hearts, Sarah receives $0.90, so the expected earnings from the ace of hearts per draw is (1/52) * $0.90 = $0.01731 (approx. $0.02).
Therefore, the total expected earnings per draw is $0.10 + $0.04 + $0.02 = $0.16.
Since the cost per draw is $0.15, Sarah's expected earnings are greater than the cost per draw. Therefore, she should play the game as she would likely make a profit on average.