I have a probability question that has to do with slot machines. Here is how the game works:
There are two reels, one on the left and one on the right. There are several symbols on each reel, one of which is an 8. The 8 symbol lands on the left reel 1 in 9 plays of the game (or spins). The 8 symbol lands on the right reel 1 in 10 plays. The reels spin independently of each other, and each play is independent from previous plays. Each play of the game spins both reels. Each reel has a large 8 above it, split into 8 segments. As the 8 symbol lands on the corresponding reel below it, one additional segment of the large 8 lights up. Once both large 8s are fully lit (i.e. at least 8-8s have landed on each reel) a 50 credit bonus is paid and the game ends. If one reel’s large 8 is already fully lit, the game isn’t over yet (i.e. do NOT pay the 10 credit bonus on a spin that completes the game), and a new 8 lands on its corresponding reel, a 10 credit bonus is paid. Each play of the game costs 1 credit.
I need to compute the expected number of credits won or lost while playing this game.
Well I have an idea on how to approach this. On the first reel, an eight appears 1 every 9 spins. On the second reel, it appears 1 every 10 spins. Therefore, the odds favoring an 8 is 1 in 9 and 1 in 10. What I think the expected value of this scenario is this: If $n$ is the number of reels, then
$$EV = -1(\frac{n-1}{n})(\frac{n-1}{n}) + 9(\frac{1}{10})(\frac{n-1}{n}) + 9(\frac{1}{11})(\frac{n-1}{n}) + 49 (\frac{1}{10})(\frac{1}{11}). $$
Do you think I am approaching in the right direction? Thank you for your help!