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.