The "at least once" rule is based on the complement rule in probability, which states that:

$P(\text{at least one success}) = 1 - P(\text{no successes})$

In this case, we want to find the probability of purchasing at least one winning lottery ticket out of 10 tickets, given that the probability of winning on a single ticket is $P(\text{win}) = 0.08$.

Step-by-step solution:

1. Calculate the probability of not winning on a single ticket: $P(\text{no win on a single ticket}) = 1 - 0.08 = 0.92$

2. Calculate the probability of not winning on all 10 tickets: $P(\text{no wins on 10 tickets}) = 0.92^{10}$ Using a calculator: $0.92^{10} \approx 0.4228$

3. Apply the complement rule to find the probability of winning at least one ticket: $P(\text{at least one win}) = 1 - P(\text{no wins on 10 tickets})$ $P(\text{at least one win}) = 1 - 0.4228 = 0.5772$

Thus, the probability of purchasing at least one winning lottery ticket out of 10 tickets is approximately 0.5772.

Final Answer:

$\boxed{0.5772}$

Would you like more details or have any questions about this?

Here are some related questions:

1. What is the probability of winning on exactly one ticket?
2. How does the probability change if you buy 20 tickets?
3. How does this rule apply if you buy fewer than 10 tickets?
4. What is the probability of winning on at least two tickets?
5. How can we express this problem using binomial probability formulas?

Tip: Remember that for "at least one" problems, calculating the complement (no successes) can simplify your calculations!