Math Problem Statement
Solution
The image depicts a probability problem where Keith is playing a game of chance with a deck of cards numbered 1 through 10. The problem involves calculating the expected value of the game based on the rules provided.
Problem summary:
- Deck composition: 10 cards numbered from 1 to 10.
- Rules:
- If Keith draws an odd-numbered card, he wins an amount of money equal to the value of the card.
- If Keith draws an even-numbered card, he loses $6.
Questions:
- (a) Find the expected value of playing the game.
- (b) What can Keith expect in the long run after playing many times?
Solution to part (a):
To find the expected value, we need to compute the sum of the probabilities multiplied by the outcomes.
Odd-numbered cards:
- Odd numbers in the deck: 1, 3, 5, 7, 9.
- Probability of drawing an odd-numbered card: .
- For each odd-numbered card drawn, Keith wins an amount equal to the card value.
The expected value for the odd-numbered cards is calculated by averaging the values of the odd cards: So, the expected gain from drawing an odd-numbered card is .
Even-numbered cards:
- Even numbers in the deck: 2, 4, 6, 8, 10.
- Probability of drawing an even-numbered card: .
- For each even-numbered card drawn, Keith loses $6.
The expected loss from drawing an even-numbered card is:
Total expected value:
Adding the expected value from odd and even draws: Thus, the expected value of playing the game is dollars.
Solution to part (b):
Since the expected value is , Keith can expect to lose $0.50 per draw in the long run.
Would you like further clarification or details on this? Here are some related questions you might explore:
- How would the expected value change if Keith's win on odd-numbered cards was doubled?
- What if the penalty for drawing an even-numbered card was reduced to $4 instead of $6?
- How does the law of large numbers relate to Keith’s long-term expectation of losing money?
- What would the expected value be if Keith only lost money on even-numbered cards above 6?
- How would the expected value change if the deck size increased to 20 cards, maintaining a similar structure?
Tip: Expected value calculations are essential in determining whether a game or investment is favorable in the long run.
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Math Problem Analysis
Mathematical Concepts
Probability
Expected Value
Games of Chance
Formulas
Expected Value (E) = Σ [Probability × Outcome]
Theorems
Law of Large Numbers
Suitable Grade Level
Grades 10-12
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