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 or0.5
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 like0.5
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.
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. $\endgroup$