Math Problem Statement
Solution
The image shows a problem where the player buys 5 chances to play a state’s daily number, with each number costing $3. If the player wins, they receive $850, but they lose the $3 spent to enter the game. The goal is to find the expected value for the player when purchasing 5 chances.
To solve this, let’s break down the steps:
-
The probability of winning:
There are 1000 possible three-digit numbers (from 000 to 999 inclusive), so the probability of winning on a single play is: -
The profit for one play:
If the player wins, they receive $850, but they paid $3 to play. So, the net profit for a win is: If they lose, the profit is negative $3, as they paid $3 to play and did not win. -
Expected value for one play:
The expected value for one play is calculated using the formula: Where: Substituting the values: So the expected value per play is approximately -$2.15. -
Expected value for 5 plays:
Since the player is purchasing 5 chances, the expected value for 5 plays is: Thus, the expected value for the player is approximately -$10.75.
Let me know if you'd like further details or have any questions!
Here are some related questions to explore:
- How would the expected value change if the player purchased more than 5 chances?
- What if the prize amount was doubled to $1700—how would that affect the expected value?
- How can the expected value be used to inform the player's strategy in this situation?
- What is the difference between the probability of winning and the expected value of the game?
- How does the number of chances purchased affect the variance of the player's winnings?
Tip: The expected value is a powerful tool to understand long-term outcomes in probability-based games. It helps quantify whether a game is favorable or not.
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Math Problem Analysis
Mathematical Concepts
Probability
Expected Value
Basic Arithmetic
Formulas
Expected Value = Σ (Probability of Outcome × Value of Outcome)
P(win) = 1/1000
Profit = Winnings - Cost
Theorems
Law of Total Expectation
Suitable Grade Level
Grades 10-12
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