Sarah draws a card from a deck of 52 cards. She receives 40 cents for a heart, 50 cents for an ace, and 90 cents for the ace of hearts. The cost of a draw is 15 cents. How much will Sarah lose if she plays this game?
She wins if
1. she draws a heart, but not the ace of heart (40 cents)
Prob = 12/52
2. she draws an ace , but not the ace of heart (50 cents)
prob = 3/52
3. she draws the ace of hearts ( 90 cents)
prob = 1/52
expected return
= (12/52)(40) + (3/52)(50) + (1/52)(90)
= 13.8 cents
so if she pays 15 cents to play the game ........
Well, it looks like Sarah is playing a pretty high-stakes game for some spare change. Let's break it down. If she draws a heart, she gets 40 cents, if she draws an ace, she gets 50 cents, and if she draws the ace of hearts, she gets a whopping 90 cents. Sounds good so far, right?
Now, we need to consider the cost of the draw, which is 15 cents. So, no matter what card she gets, Sarah loses 15 cents right off the bat. So, we need to subtract that from whatever she wins.
Let's see, the most she can win is 90 cents, but she needs to subtract the 15 cents cost of the draw. So, she's left with 75 cents. If she draws an ace, she wins 50 cents, but again, she needs to subtract the 15 cents for the draw, leaving her with 35 cents. If she draws any other heart, she wins 40 cents but loses 15 cents, leaving her with 25 cents.
Okay, let's sum it up. If Sarah draws the ace of hearts, she wins 75 cents. If she draws any other ace, she wins 35 cents. And if she draws any other heart, she wins 25 cents. But keep in mind, she loses 15 cents for each draw.
So, to find out how much Sarah will lose, we need to calculate the expected value. Which, in this case, is (75 * 1/52) + (35 * 3/52) + (25 * 13/52) - 15. After crunching the numbers, it seems like Sarah will lose approximately 7 cents per card draw.
Now, that may not sound like much, but hey, every penny counts, right? So, Sarah might want to think twice about playing this game unless she wants to experience the thrill of losing a few cents along the way.
To calculate how much Sarah will lose if she plays this game, we need to determine her total earnings and subtract the cost of the draw. Let's break it down step-by-step:
Step 1: Determine the earnings for each possible outcome.
- Sarah receives 40 cents for drawing a heart.
- Sarah receives 50 cents for drawing an ace.
- Sarah receives 90 cents for drawing the ace of hearts.
- The probability of drawing a heart is 13/52 (as there are 13 hearts in a deck of 52 cards).
- The probability of drawing an ace is 4/52 (as there are 4 aces in a deck of 52 cards).
- The probability of drawing the ace of hearts is 1/52 (as there is only one ace of hearts in a deck of 52 cards).
Step 2: Calculate the expected earnings for each outcome.
- Earnings from drawing a heart: 40 cents × (13/52) = 10 cents.
- Earnings from drawing an ace: 50 cents × (4/52) = 3.846 cents (rounded to 3 decimal places).
- Earnings from drawing the ace of hearts: 90 cents × (1/52) = 1.731 cents (rounded to 3 decimal places).
Step 3: Calculate the expected total earnings.
- Total expected earnings = Earnings from drawing a heart + Earnings from drawing an ace + Earnings from drawing the ace of hearts.
- Total expected earnings = 10 cents + 3.846 cents + 1.731 cents = 15.577 cents (rounded to 3 decimal places).
Step 4: Subtract the cost of the draw.
- Total expected earnings - Cost of draw = 15.577 cents - 15 cents = 0.577 cents.
Therefore, Sarah is expected to lose approximately 0.577 cents if she plays this game.
To calculate how much Sarah will lose if she plays this game, we need to calculate the expected value by taking into consideration the probabilities of drawing each type of card and the associated winnings or losses.
There are 52 cards in a deck, and since Sarah is drawing one card, the probability of drawing any particular card is 1/52.
First, let's consider the winnings or losses for each type of card:
- For a heart, Sarah receives 40 cents.
- For an ace, Sarah receives 50 cents.
- For the ace of hearts, Sarah receives 90 cents.
Now, let's calculate the probabilities:
- There are 13 hearts in a deck of 52 cards, so the probability of drawing a heart is 13/52 = 1/4.
- There are 4 aces in a deck of 52 cards, so the probability of drawing an ace is 4/52 = 1/13.
- There is only one ace of hearts in a deck of 52 cards, so the probability of drawing the ace of hearts is 1/52.
Next, let's calculate the expected value for each type of card:
- Expected value for a heart: (40 cents) * (1/4) = 10 cents.
- Expected value for an ace: (50 cents) * (1/13) = 3.846 cents (rounded to 3 decimal places).
- Expected value for the ace of hearts: (90 cents) * (1/52) = 1.731 cents (rounded to 3 decimal places).
Now, let's calculate the overall expected value:
- The cost of a draw is 15 cents, so the expected value of losing is -15 cents.
- The overall expected value is given by the sum of the expected values for each type of card and the expected value of losing: 10 cents + 3.846 cents + 1.731 cents - 15 cents = 0.577 cents (rounded to 3 decimal places).
Since the overall expected value is positive, it means that on average, Sarah can expect to win 0.577 cents per game. Therefore, Sarah will lose approximately 0.577 cents if she plays this game.