# Expected Win (or Loss) of Base (Spin) Game and Bonus (Guess the Colour) Game

I'm trying to figure out the right way of calculating an expected win (or loss) for a slot base game spin with a bonus game (guess the colour).

Let's say a player makes a $1 bet and gets a combination that pays$10 (еру base game). The player can either take their winnings ($10) or play the bonus game. In the bonus game the player has to guess the colour (red or black). If the player guesses it right, the winnings will be doubled (2 ×$10 = $20). Otherwise, they lose the winnings ($0) and the bonus game stops. Again, the player can either take their winnings ($20) or play the next round of the bonus game. There are maximum of 5 bonus game rounds allowed. Thus, there are the following possible outcomes: • \$10 (the player decides not to play the bonus game. Just takes the prize they win in the base game);
• \$20, \$40, \$80, \$160, or \$320 (the player plays 1, 2, 3, 4, or 5 rounds of the bonus game and takes that prize); • \$0 (the player plays 1, 2, 3, 4, or 5 rounds of the bonus game and loses).

Here is what I have tried so far:

For the win/loss table, I think the initial bet of $1 should be used. Or, maybe, the potential prize from the previous round/game should be listed in the Loss column? Game Round Win, \$ Loss, \$Base - 10 0 Bonus 1 20 1 Bonus 2 40 1 Bonus 3 80 1 Bonus 4 160 1 Bonus 5 320 1 Probability of loss is $$p_{loss} = 1 - p_{win}$$. But I'm not sure what probabilities of win should be used: • 0.5 for all cases or • 0.5 for the base game (probability the player will decide to play the bonus game) and $$0.5^{round}$$ for the bonus games (probability of getting all played rounds won) or • something else, since not all combinations are possible (e.g., for 3 rounds, combination win, loss, win is impossible, since the bonus game ends after the loss). This looks like 0.5 should be used for all cases. Game Round pwin ploss Base - 0.5 0.5 Bonus 1 0.51 = 0.5 1 – 0.51 = 0.5 Bonus 2 0.52 = 0.25 1 – 0.52 = 0.75 Bonus 3 0.53 = 0.125 1 – 0.53 = 0.875 Bonus 4 0.54 = 0.0625 1 – 0.54 = 0.9375 Bonus 5 0.55 = 0.03125 1 – 0.55 = 0.96875 Then, I think, the average of the expected values for the base and bonus games should be calculated as followsNOTE: $$E[game] = \frac{E[spin]+\sum_{i=1}^5 E[round_i]}{n_{spins}+n_{rounds}} = \frac{(p_{win}\times win-p_{loss}\times loss)_{spin}+\sum_{i=1}^5 (p_{win}\times win-p_{loss}\times loss)_{i}}{1+5}$$ So, the question is: what's the correct way to calculate the expected win for this case? NOTE I'm not 100% sure that the formula is written correctly in terms of notation, since I'm not very familiar with the probability and statistics. • You'd realistically lose money, as a 1 dollar entry fee with a 50% of winning 10 dollars, means that everyone who only plays (only) the first round would make on average 4 dollars. And a lot of people would figure that out, (you should probably raise the entry fee) – Sejr May 29 at 13:44 • Sejr, thank you for your comment, but I want to figure out how to calculate that. May 29 at 13:50 • I think I understand the quistion better now, so is it this? ( where B_win is the winning chance on base round, which seems to be different than the other bonus rounds. And P_stop is the chance of the player stopping every round ) (The player on average earns around 3.42):$\frac{1}{1-P_{stop}^{1+R_{rounds}}}\left(\left(P_{stop}\right)\left(\left(B_{win}\right)\left(E_{earning}\right)-F_{fee}\right)+\left(B_{win}\right)\sum_{n=1}^{R_{rounds}}\left(P_{stop}^{1+n}\left(\left(P_{win}\right)^{\left(\frac{n^{2}+n}{2}\right)}\cdot\left(2\right)^{n}\cdot E_{earning}-F_{fee}\right)\right)\right)$– Sejr May 29 at 14:49 • -So you want an entry fee on 5.53 or higher, if you want to earn money. – Sejr May 29 at 14:51 • Sejr, correct me, if I'm wrong. The bonus game can be triggered only if the base game gives a win. Thus,$ B_{win} = 1 \$. But there is a chance that the player will not play the bonus game (I believe it is 0.5). I don't know what entry fee you mean. And I don't need to calculate it. I just need to calculate the expected win or loss with the given bet and prizes. May 29 at 15:14

$$\text{Average Earning after 1 game}=-\frac{1}{1-P_{stop}^{1+R_{rounds}}}\left(\left(P_{stop}\right)\left(\left(B_{win}\right)\left(E_{earning}\right)-F_{fee}\right)+\\ \left(B_{win}\right)\sum_{n=1}^{R_{rounds}}\left(P_{stop}^{1+n}\left(\left(P_{win}\right)^{\left(\frac{n^{2}+n}{2}\right)}\cdot\left(2\right)^{n}\cdot E_{earning}-F_{fee}\right)\right)\right)$$
(Where $$B_{win}$$ is the winning chance on base round, which seems to be different than the other bonus rounds. And $$P_{stop}$$ is the chance of the player stopping every round, the rest should be self explanatory )