In a​ state's lottery, you can bet ​$4 by selecting three​ digits, each between 0 and 9 inclusive. If the same three numbers are drawn in the same​ order, you win and collect ​$300.

To analyze the lottery game described, we need to calculate the probability of winning and the expected value of a single bet.

### 1. Probability of Winning

In this lottery, you select three digits (each between 0 and 9), and the order in which the digits are selected matters.

- There are 10 choices for each digit (0 to 9).
- The total number of possible combinations is:
  
\[
10 \times 10 \times 10 = 1,000.
\]

Since only one specific combination will match the drawn numbers in the exact order, the probability of winning is:

\[
P(\text{winning}) = \frac{1}{1,000} = 0.001.
\]

### 2. Expected Value of a Single Bet

The expected value (EV) of a bet is calculated by considering all possible outcomes, their probabilities, and their associated payoffs.

- **Winning Outcome:** 
  - You win $300, but since you paid $4 to bet, your net gain is $300 - $4 = $296.
  - Probability of winning: $P(\text{winning}) = 0.001$.
  - Contribution to expected value from winning: $0.001 \times 296 = 0.296$.

- **Losing Outcome:**
  - You lose your $4 bet.
  - Probability of losing: $P(\text{losing}) = 1 - P(\text{winning}) = 0.999$.
  - Contribution to expected value from losing: $0.999 \times (-4) = -3.996$.

Now, the expected value of a single bet is:

\[
\text{EV} = (0.001 \times 296) + (0.999 \times (-4))
\]

\[
\text{EV} = 0.296 - 3.996
\]

\[
\text{EV} = -3.70.
\]

### Conclusion

The expected value of a single bet in this lottery game is **-$3.70**. This means that, on average, you lose $3.70 for every $4 bet you place.

Would you like more details or have any questions?

### Related Questions:
1. What is the probability of not winning in the lottery?
2. How would the expected value change if the payout increased to $500?
3. What is the break-even point in this lottery game?
4. How does the probability of winning change if you can select four digits instead of three?
5. What is the variance in the outcomes of this lottery game?

### Tip:
When considering gambling or lottery games, it's important to understand the concept of expected value to gauge whether the game is fair or profitable over the long run.

Explore the probability and expected value analysis of a state lottery where a $4 bet can win $300 if three selected numbers match the drawn numbers in exact order. Understand the mathematical calculations involved and the implications for betting strategies.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation