Question
Amanda has a deck of 10 cards numbered 1 through 10 . She is playing a game of chance.
This game is this: Amanda chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an odd numbered ca
drawn. She loses
(a) Find the expected value of playing the game.
dollars
(b) What can Amanda expect in the long run, after playing the game many times?
(She replaces the card in the deck each time.)
She can expect to win
Amanda can expect to lose money.
She can expect to lose
Amanda can expect to break even (neither gain nor lose money).
This game is this: Amanda chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an odd numbered ca
drawn. She loses
(a) Find the expected value of playing the game.
dollars
(b) What can Amanda expect in the long run, after playing the game many times?
(She replaces the card in the deck each time.)
She can expect to win
Amanda can expect to lose money.
She can expect to lose
Amanda can expect to break even (neither gain nor lose money).
Ask by Adkins Perkins. in the United States
Jan 06,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a) The expected value of playing the game is –$0.50.
(b) Amanda can expect to lose $0.50 per draw in the long run.
Solution
Let’s analyze Amanda’s game step by step.
Part (a): Calculating the Expected Value
Given:
- Total cards: 10 (numbered from 1 to 10)
- Odd-numbered cards: 1, 3, 5, 7, 9
- Even-numbered cards: 2, 4, 6, 8, 10
- Probability of drawing any specific card:
- Winning condition: If an odd card is drawn, Amanda wins an amount equal to the card’s value.
- Losing condition: If an even card is drawn, Amanda loses $6.
Step-by-Step Calculation:
-
Calculate the total value from odd-numbered cards:
-
Calculate the total loss from even-numbered cards:
-
Compute the expected value (E):So, the expected value is -$0.50.
Part (b): Long-Run Expectation
Based on the expected value calculated:
- Expected Value per Draw: -$0.50
Interpretation:
- Since the expected value is negative, Amanda can expect to lose money in the long run.
- Specifically, she can expect to lose $0.50 per draw on average.
Final Answers:
(a) The expected value of playing the game is – $0.50.
(b) Amanda can expect to lose $0.50 per draw in the long run.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Explain 
Simplify this solution Beyond the Answer
To calculate the expected value of playing Amanda’s game, we first need to understand the outcomes based on the cards drawn.
(a) To find the expected value, we consider the winnings and losses for each card:
- Odd cards: 1, 3, 5, 7, 9 (Total winnings:
) - Even cards: 2, 4, 6, 8, 10 (Total loss per each is -$6)
Each card has an equal probability of
Now we calculate the expected winnings from odd cards:
Total probability for odd draws = 5 cards =
Expected loss from even cards:
Total probability for even draws = 5 cards =
Combining everything, the expected value (EV) can be calculated as follows:
So the expected value of playing the game is
(b) In the long run, after playing the game many times, Amanda can expect to lose money. Since the expected value is negative, she can expect to lose
