Free Spins Math¶
Trigger probability¶
For each base reel, the calculation finds the probability of 0, 1 and 2 FS symbols in the window. The reels are independent. A convolution of the 5 reel distributions gives the distribution of the total.
| FS symbols in the window | Probability |
|---|---|
| 0 | 0.63042580 |
| 1 | 0.30524570 |
| 2 | 0.05850317 |
| 3 | 0.00555827 |
| 4 | 0.00026214 |
| 5 | 0.00000492 |
| Trigger | Probability | Frequency | Initial free spins |
|---|---|---|---|
| 3 FS | 0.00555827 | 1 in 179.9 | 10 |
| 4 FS | 0.00026214 | 1 in 3,814.7 | 15 |
| 5 FS | 0.00000492 | 1 in 203,450.5 | 20 |
| Total | 0.00582533 | 1 in 171.66 |
KEY symbols in one free spin¶
The same method gives the number of KEY symbols in one free spin on the bonus reel strips.
| KEY symbols in one free spin | Probability |
|---|---|
| 0 | 0.812544 |
| 1 | 0.175168 |
| 2 | 0.012032 |
| 3 | 0.000256 |
The mean is 0.2000 KEY symbols for each free spin.
Expected value of one free spin¶
The ways pays and the collect use the formulas of the base game, with the bonus reel strips. The collect uses the weight tables of the level.
If multiplierAppliesTo = all:
spin EV(level) = multiplier(level) x (EV ways + EV collect(level))
If multiplierAppliesTo = fish:
spin EV(level) = EV ways + multiplier(level) x EV collect(level)
This configuration uses all.
| Level | Multiplier | EV ways (credits) | EV collect (credits) | Spin EV (credits) | Spin EV (x bet) |
|---|---|---|---|---|---|
| 1 | x1 | 3.100092 | 10.678853 | 13.778945 | 1.3779 |
| 2 | x2 | 3.100092 | 11.085973 | 28.372131 | 2.8372 |
| 3 | x5 | 3.100092 | 11.354645 | 72.273687 | 7.2274 |
| 4 | x10 | 3.100092 | 11.652133 | 147.522254 | 14.7522 |
Level probabilities¶
The state of the free spins is the total number of KEY symbols, k.
The calculation holds the probability of each state. The top state is "9 or more".
- Level 1 starts with probability 1 in state
k = 0. - For each spin, the calculation moves the probability from state
kto statek + j. The factor is the probability ofjKEY symbols in one spin. - At the end of a level, the states below the threshold of the next level stop. Their probability leaves the calculation.
- The sum of the probabilities that stay is the probability to play the next level.
- The calculation does steps 2 to 4 again for each level.
P(play level 1) = 1
P(play level L) = sum of the state probabilities with k >= threshold(L), at the end of level L-1
The win of a spin in level L is independent of the event "the player gets to level L".
That event depends only on the spins before level L. Thus the expected values multiply:
EV(bonus | trigger t) = sum for all levels L of P(play level L | t) x spins(L, t) x spin EV(L)
EV(bonus per base spin) = sum for all triggers t of P(t) x EV(bonus | t)
| Trigger | Initial spins | P(play level 1) | P(play level 2) | P(play level 3) | P(play level 4) | Mean free spins |
|---|---|---|---|---|---|---|
| 3 FS | 10 | 1.000000 | 0.323171 | 0.122812 | 0.051681 | 13.981 |
| 4 FS | 15 | 1.000000 | 0.585191 | 0.292414 | 0.143597 | 23.170 |
| 5 FS | 20 | 1.000000 | 0.772782 | 0.480314 | 0.273822 | 32.215 |
Results¶
The table shows the expected win of each level as a multiple of the bet.
| Trigger | Probability | Level 1 (x bet) | Level 2 (x bet) | Level 3 (x bet) | Level 4 (x bet) | Bonus EV (x bet) |
|---|---|---|---|---|---|---|
| 3 FS | 0.00555827 | 13.779 | 7.335 | 7.101 | 6.099 | 34.314 |
| 4 FS | 0.00026214 | 20.668 | 13.283 | 16.907 | 16.947 | 67.805 |
| 5 FS | 0.00000492 | 27.558 | 17.540 | 27.771 | 32.316 | 105.185 |
| Figure | Value |
|---|---|
| Mean free spins win for each trigger | 35.8812 x bet |
| Free spins EV for each base spin | 2.090199 credits |
| Free spins RTP | 20.9020% |
Largest wins¶
The game has no win limit. The largest collect is possible on level 4, with a KEY symbol on each of the 3 bonus reels that have KEY symbols, the FISH4 symbol and the largest FISH values. The largest round win in the simulation of 1,000,000,000,000 rounds was 10,498.0 x bet.