[resolved] Probability of x within n attempts - two answers

[Matt_T_hat]

I've been looking at a "system" someone suggested for a particular roulette table and so wanted to test their claims a little more closely. The "system" says that if you bet red with $1 and then double until you win you should increase the pot by $1 a time and always be better off.

The table in questions has 18 reds, 18 blacks and two greens (0 and 00).

That gives us 38 possible outcomes.

(opps)
P(Red) = 18/32
P(Not RED) = 20/32
(/opps)

P(Red) = 18/38
P(Not RED) = 20/38

So far so good.

What I am looking for is

P(Red in 8 or less tries)

Now if I take the probability of NOT RED 8 times I have (20/32)^8 and could assert that 1 - (20/32)^8 = P(Red With 8 tries) = 0.9941

which makes the system look good.

However, something bugged me about that it seemed to be TGTBT (too good to be true) so I ran a different set of numbers. In my head I read off all the ways a win could happen probability tree style.

(oops /)

18/38 - win
20/38 * 18/38 - win
20/38 * 20/38 * 18/38 - win
20/38 * 20/38 * 20/38 * 18/38 - win

= 0.9376

Answer one gives a positive expectation over a large number of rounds of the system but answer two gives a negative expectation over a large number of rounds.

My rusty memory of probability says that the second calculations are correct and the house still wins but I am not sufficiently full of myself these days to state this with 100% certainty.

Would someone else check my working and comment?