Probability of net gain given 100 pulls of slot machine with 3 slots, each numbered 1 through 9, and 1 dollar per push with average winning for any given winning push equal to 2 dollars (happening when there are either 2 of the same number, or 3 of the same number)?
First I considered the probability of getting either 2 of the same number, or 3 of the same number. I came up with P(two 8s) = (3C2)(1/9)^2(8/9) = 24/729. And, P(three 8s) = (3C3)(1/9)^3 = 1/729. So, P(two 8s, or three 8s) = 25/729 = 3.4%
Then considering 100 pulls of the slot machine, I got (100 C 50)(3.4%)^50(96.6%)^50 which came up to be a very small number, which I do not think is right. I considered winning 50 times with 2 dollars means you gained 100 dollars, and put in 100 dollars. So this probability, I think, represents the probability of breaking even. Is this correct? If so, how to compute probability of a net gain?