Slot problem

For simplicity, we will make it a two reel game consisting of a base game and a bonus game. The base game is the outcome of the two reels, and the bonus game is the outcome of a game played on a video display above the reels. In every base game there are a number of different symbols on the reels, one of which is a 7. There is a 7 on each reel. The probability of a 7 appearing on a spin on reel n is denoted P(7 on reel n). The outcome of each reel is picked independent of the other reel.

The bonus game consists of a “7” in the video display over each reel. The “7” in the video display is composed of seven pieces. Each time a 7 lands on the reels below, a piece of the corresponding “7” in the video display is filled in. The top award is won when both puzzle 7’s in the video display are filled. At this time the player is paid 200 coins and the puzzle is cleared. When a 7 appears on a base game reel and the corresponding “7” in the video display is already complete, the player is given a 2 coin bonus.

Find the expected number of coins won or lost upon completion of both 7’s given:
1. The player makes a constant wager of two coins per spin.
2. The two reels are independent.
3. Spins are independent. Any one spin does not depend on a previous spin.
4. P(7 on reel 1) = 1/24
P(7 on reel 2) = 1/36
5. The 2 coin bonus is not paid for a spin that completes the video 7 for a reel.
6. The 2 coin bonus is not paid for a spin that completes the entire puzzle (both 7’s) even if a 7 lands on a reel that already has the corresponding 7 puzzle lit.

I can't find a solution without oversimplifying it, i.e. under the assumption that reel 1 always completes first, it'd take 36*7 = 252 spins for reel 2 to complete, and in that time, reel 1 would have paid out 3 times, for a total of 6 coins, the net coin result would be 200 + 6 - 252*2 = -298.

Is this a game you are designing?

No, just a problem that was posed to me by a friend. Can't figure out the math behind it and decided to seek help from the interwebs!

I was able to solve this with a Markov Matrix. On average, with will take just over 265 games to complete both puzzles and will have about 4.4 extra hits while waiting to solve the second puzzle. The return would be approximately 208.8/530. I have exact numbers, but this seems good to satisfy curiosity.

Good call on the Markov chains, will work on it tomorrow and I'll see what I come up with, my simulations have been around a -320, so that sounds about right.