Does probability change is the denominator is split into multiple pools or concentrated into a single pool?

Question

Having a debate and would love a statistically sound confirmation :) Here is the scenario:

You have 9 tickets to enter into a drawing pool. That pool of individual entries is 100 each day. You have a choice of entering all 9 in a single day (9/100 = 9%) or you can choose to enter them over 3 different days of pools with a size of 100, 3 tickets each day. 3/100 * 3 days = 9%.

Are these in fact the same odds and probability?

It feels like condensing them into the same pool would be better (100) vs going up against 3x pools of 100 (300).

It’s a very competitive debate so scientific mathematical evidence of which is better (or are they the same) would be VERY AWESOME. Thanks!

Answer 1

Your expected number of wins is the same in either case at $0.09$. The probability of at least one win is slightly higher if you enter all of them the same day. It is also $0.09$. If you enter three each on three days there is some chance that you will win two or three prizes, so the chance of at least one is reduced. In this case the chance of not winning on a day you enter is $0.97$, so the chance of losing every time is $0.97^3$ and the chance of at least one in is $1-0.97^3 =0.087327$ You need to be careful what question you are asking.

Answer 2

When you enter all in one day the chance of losing is $0.91$. If you enter 3 each in 3 days, the chance of losing all is $0.97^3=0.912673$, so you are slightly better off if you enter all in one day.

However if you enter in 3 days, there is a chance you will win more than once.