Odds of lottery if more than 1 grand prize

Question

There are 60,000,000 possible combinations for a lottery, and there are 3 combinations that will win the "Grand Prize".

So the odds of winning the grand prize is 3/60,000,000 = 1 in 20,000,000.

1. How would I calculate the odds to win at least 1 of the 3 grand prizes? If I buy 10 tickets, is my odds of winning (at least) 1 grand prize 10/60,000,000 or 10/20,000,000 ?

2. If I bought 20,000,000 tickets, does that mean my chance to winning at least 1 grand prize is 20,000,000/60,000,000 = 1/3?

Answer 1

Some hints:

• If you purchase a given number of tickets, you either get none of the grand prizes, or you get at least one of the grand prizes. These two events, taken together, have probability $1$.
• The probability of not getting a grand prize on the first ticket is $59,999,997/60,000,000$.
• The probability of not getting a grand prize on the second ticket, assuming you didn't get it on the first ticket, is $59,999,996/59,999,999$.

Can you take it from here?