A local lottery draws one winner out of 1,000,000 tickets. You see two winners
interviewed on TV this week, which makes you think that your chance of winning has
increased. You plan to buy 10 tickets this week.
 What is the exact probability that you will win with one ticket, and what is the
probability that you will win if you buy 10 tickets?

Let's break down the problem and find the probabilities step by step.

### Probability of Winning with One Ticket:
The total number of tickets in the lottery is 1,000,000. The probability of winning with one ticket is simply the chance of drawing the winning ticket from these 1,000,000 tickets.

The probability of winning with one ticket is:
\[
P(\text{win with 1 ticket}) = \frac{1}{1,000,000}
\]

### Probability of Winning with 10 Tickets:
If you buy 10 tickets, the probability of winning with at least one of those tickets is a little more complex to calculate. To determine this probability, we can use the complement rule, which says:

- First, calculate the probability that you **don’t win** with any of the 10 tickets.
- Then subtract this from 1 to find the probability that you win with at least one ticket.

The probability of **not winning** with one ticket is:
\[
P(\text{not winning with 1 ticket}) = 1 - \frac{1}{1,000,000} = \frac{999,999}{1,000,000}
\]

The probability of **not winning** with all 10 tickets is:
\[
P(\text{not winning with 10 tickets}) = \left(\frac{999,999}{1,000,000}\right)^{10}
\]

Thus, the probability of winning with at least one of the 10 tickets is:
\[
P(\text{win with 10 tickets}) = 1 - \left(\frac{999,999}{1,000,000}\right)^{10}
\]

Now, let’s compute the values:

1. **Probability of winning with one ticket:**
\[
P(\text{win with 1 ticket}) = \frac{1}{1,000,000} = 0.000001
\]

2. **Probability of winning with 10 tickets:**

First, we calculate:
\[
\left(\frac{999,999}{1,000,000}\right)^{10} \approx 0.99999^{10} \approx 0.99999
\]
Now, subtract this from 1:
\[
P(\text{win with 10 tickets}) = 1 - 0.99999 = 0.00001
\]

So, the exact probability of winning with 10 tickets is approximately 0.00001, or 0.001%, which is still very small.

### Summary:
- The probability of winning with **one ticket** is **0.000001** (or **0.0001%**).
- The probability of winning with **10 tickets** is **0.00001** (or **0.001%**).

Would you like more details on the complement rule or any other part of the calculation?

### Related Questions:
1. How does the probability of winning change if you buy 100 tickets?
2. What is the probability of losing with 10 tickets?
3. How does buying additional tickets affect your chances in other lotteries?
4. Can the probability of winning be exactly 1 with a certain number of tickets?
5. How do you calculate the probability of at least one win with multiple independent events?

### Tip:
When calculating the probability of multiple independent events (like drawing winning tickets), the complement rule is a useful tool for simplifying the process.

A local lottery draws one winner out of 1,000,000 tickets. You see two winners interviewed on TV this week, which makes you think that your chance of winning has increased. You plan to buy 10 tickets this week. What is the exact probability that you will win with one ticket, and what is the probability that you will win if you buy 10 tickets?

Learn how to calculate the probability of winning in a lottery with one or more tickets. This problem explains the use of the complement rule and provides step-by-step calculations for both a single ticket and ten tickets. Ideal for students learning probability.

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