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

Question

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 follows^NOTE:

$$ E[game] = \frac{E[spin]+\sum_{i=1}^5 E[round_i]}{n_{spins}+n_{rounds}} = \frac{(p_{win}\times win-p_{loss}\times loss)_{spin}+\sum_{i=1}^5 (p_{win}\times win-p_{loss}\times loss)_{i}}{1+5} $$

So, the question is: what's the correct way to calculate the expected win for this case?

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^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.

Answer 1

I'm fairly sure that the equation below is what you are lookin for, I would recommend playing around with it in Desmos (https://www.desmos.com/calculator/sq5gkraui7)

$\text{Average Earning after 1 game}=-\frac{1}{1-P_{stop}^{1+R_{rounds}}}\left(\left(P_{stop}\right)\left(\left(B_{win}\right)\left(E_{earning}\right)-F_{fee}\right)+\\ \left(B_{win}\right)\sum_{n=1}^{R_{rounds}}\left(P_{stop}^{1+n}\left(\left(P_{win}\right)^{\left(\frac{n^{2}+n}{2}\right)}\cdot\left(2\right)^{n}\cdot E_{earning}-F_{fee}\right)\right)\right)$

(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 )

The player on average earns around 3.42, and if you want to make money, the entry fee has to be around 5.53 or higher.