I got involved in a debate with my sister this evening, and despite (what I thought) was a relatively simple problem, I couldn't provide a rigorous answer. I was hoping that someone on here could give me some insight.

A casino she frequents will often give her "fake money" to use at a spinning wheel. For the purpose of this question, let's assume the wheel is a fair one and it has 10 spaces numbered 1-10. If she puts her "fake money" on any slot, and it comes up on the spin, she gets REAL money for it.

Now, let's assume she has 10 "fake" dollar bills. If she puts a single dollar on EVERY space, she is guaranteed to win $1 in real money. However, if she puts $1 on the *SAME* space for 10 spins, she is NOT guaranteed to win even though the probability of winning on each spin is 1/10.

If we spin 10 times, and the probability of a win is 1/10, shouldn't:

1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + = 1

I'm confused as to why mathematically these "independent" events don't add up to certainty the way that placing bets on EVERY space does.

I know this is a trivial question to many of you, but it has been driving me nuts.

Thanks so much,

Ben