Raffle Probability

Question

I'm trying to figure out my odds are winning a raffle. I'm simplifying it here because clearly odds won't be as good as the example, but it will be easier for me to understand.

There are 100 numbers and they will choose 5 numbers to win. If you buy more tickets, you're allowed to win multiple times. Now, if I buy 10 tickets, what are my chances of winning at least once?

Here's some of my thinking...

1) 5% chance of winning (5 winners / 100 numbers) if I have 1 ticket.

2) 50% chance of winning (5% chance of winning per ticket x 10 tickets) if I have 10 tickets.

3) # 2 doesn't make sense since I have 10/100 tickets = 10%. I don't think it's additive. What am I missing?

Answer 1

This probability example is easier to solve if you calculate the chance of not winning and take the opposite. The wording is a confusing, but this is what I mean:

If you have 1 ticket: The chance you don't win on the first draw is 99/100. The chance you don't win on the second draw is 98/99, third 97/98,... The chance you don't win at all is these 5 scenarios multiplied by each other, which will equal .95. The chance you win with that one ticket would be (1 - .95) or .05.

The same applies if you have 10 tickets: The chance you don't win on the first draw is 90/100. The chance you don't win on the second draw is 89/99, third 88/98,...The chance you don't win at all is these 5 scenarios multiplied by each other, which will equal approximately .584. The chance you win atleast once, would be approximately (1 - .584) or .416.