Let A denote the event of placing a ​$1 straight bet on a certain lottery and winning. Suppose​ that, for this particular​ lottery, there are 3 comma 456 different ways that you can select the four digits​ (with repetition​ allowed) in this​ lottery, and only one of those​ four-digit numbers will be the winner. What is the value of​ P(A)? What is the value of ​P(Upper A overbar​)?

We are given the following information:

- There are 3,456 different ways to select the four digits in the lottery (with repetition allowed).
- Only one of these four-digit numbers will win.
- We need to calculate the probabilities $ P(A) $ and $ P(\overline{A}) $, where:
  - $ A $ is the event of winning the lottery (i.e., placing a $1 bet and winning).
  - $ \overline{A} $ is the complement of event $ A $, meaning the event of not winning the lottery.

### Step 1: Probability of winning, $ P(A) $

Since only 1 out of the 3,456 possible selections is a winning number, the probability of winning the lottery is the ratio of the favorable outcomes (1) to the total possible outcomes (3,456).

\[
P(A) = \frac{1}{3,456}
\]

### Step 2: Probability of not winning, $ P(\overline{A}) $

The probability of not winning the lottery is the complement of the probability of winning, which is calculated as:

\[
P(\overline{A}) = 1 - P(A)
\]

Now let's calculate these values.

#### Calculation:

\[
P(A) = \frac{1}{3,456} \approx 0.000289
\]

\[
P(\overline{A}) = 1 - \frac{1}{3,456} = 1 - 0.000289 \approx 0.999711
\]

### Conclusion:

- The probability of winning, $ P(A) $, is approximately 0.000289.
- The probability of not winning, $ P(\overline{A}) $, is approximately 0.999711.

Would you like more details on how these probabilities were derived?

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Here are 5 related questions:
1. How does the probability of winning change if the number of possible outcomes increases?
2. What is the probability of winning twice in a row?
3. What is the expected number of bets before winning the lottery?
4. How would the probabilities change if the lottery system allowed fewer digits to be selected?
5. What would the probability of winning be if you selected two different combinations of numbers?

**Tip**: Complementary probabilities like $ P(\overline{A}) $ are often easier to compute than the direct event, especially when the event has a very low probability.

Let A denote the event of placing a ​$1 straight bet on a certain lottery and winning. Suppose​ that, for this particular​ lottery, there are 3,456 different ways that you can select the four digits​ (with repetition​ allowed) in this​ lottery, and only one of those​ four-digit numbers will be the winner. What is the value of​ P(A)? What is the value of ​P(Upper A overbar​)?

Explore the probability of winning a lottery where there are 3,456 possible four-digit combinations. This problem covers key concepts in probability theory, including complementary events. Detailed step-by-step solutions are provided, making it ideal for students in Grades 9-12.

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