Andrew, I don't think I'm following your logic here. The ratio of 300/9000 does reduce to 1/30, so that's the number of winning tickets out of the total number of tickets, at least at the start. But what does x/5 mean, and how does that help answer the probability of winning?
Here's how I'd think about it:
The question asks for the probability of picking a winner. But that's complicated, because I could pick one winner and four losers, or two winners with three losers, etc etc etc...heck, if I'm lucky, all 5 are winners, and I didn't pick any losers at all. So, figuring out those probabilities is a pain.
Rather than go at it directly, instead of counting the ways of picking a winner, how about we look at the alternative: not picking any winners at all. What if all 5 tickets are losers...
On Ticket #1, I have a probability of 8700/9000 of picking a loser.
Once I've done that, on ticket #2, there are 8699 losing tickets left out of the remaining 8999 tickets. So now, I've got a 8699/8999 chance of picking a loser.
On ticket #3, I'm down to 8698 possible losers of the 8998 tickets remaining.
And so on. Ticket #4 is a loser 8697/8997, and Ticket #5 is a loser 8696/8996.
The probability of doing all five of those in sequence, ticket #1 losing and ticket #2 losing and ticket #3 losing...would be all of those probabilities multiplied together. That's the probability that I don't win on any of my five picks. I think that comes out to .844, or 84.4%, but someone can please check my math on that.
If my probability of losing is 84.4%, then the rest of the time, I'd win. So, that's a probability of 0.156, or 15.6%.
Michael W.
01/04/17