How to calculate the odds of winning a bonus round

The bonus round is a set of rules on top of the base game: freespins, multipliers, sticky wilds, collectors, wheel of prizes, "hold & spin" with respins and accumulation. To calculate chances, you need to turn mechanics into a probabilistic model, determine the event "success" and calculate probability and expectation.

1) Formalizing bonus mechanics

1. Bonus type:

Frispins with a fixed number of spins (N) and multipliers.
Hold & Spin/Respins: start with (K) cells and 3 respins; each new character resets the counter to 3.
Wheel/trail: discrete segments/steps with known odds.
2. Winning unit: multiplier to bet (X) per round.
3. Threshold of "significant success": for example, (X\ge t) (≥×10, ≥×50, etc.).
4. What is accidental: dropping characters, multipliers, adding spins, triggering upgrades.

2) Choosing a model for mechanics

A) Frispins without complex chains

If each spin is independent and the multiplier (M) is fixed, then

[
X=\sum_{i=1}^{N} M\cdot Y_i, ]

where (Y_i) is the spin win multiplier (0, 0. 2, 1, 5, …). Then:

(\mathbb{E}[X]=N\cdot M\cdot \mathbb{E}[Y])
(\mathrm{Var}(X)=N\cdot M^2\cdot \mathrm{Var}(Y))

B) Frispins with "sticky" wilds/hoarding

The state of the back depends on the past (how many wilds have already stuck). Markov chain is suitable: state = configuration of wilds/multiplier, transitions with their probabilities, and reward is the expected gain in state. Total expectation is the sum of the expected rewards by steps.

В) Hold & Spin / “coin feature”

Respins continue while new coins appear in the window (S). Denote by (p) - the probability of "catching at least one coin in the respin." Then the number of respins before stopping has a distribution with the parameter "success = zero coins"; the chances of filling all (S) cells and the average number of collected coins are calculated through geometry/binomial and recursion (below is a simplified scheme).

D) Wheel/Trail

Outcome tree: in nodes - segment probabilities, in leaves - rewards. The probability of the event (X\ge t) is the sum of the probabilities of all leaves with the payment of ≥ (t). Wait - Amount (p_\ell\cdot x_\ell).

3) Base quantities you need

Frequency of the result per spin (for freespins): (q_k=\mathbb{P} (Y = k)) or baskets (0; ≤×1; × 1- × 5; ≥×5).

Probability of triggering bonus gains (spin addition, multiplier upgrade).

For Hold & Spin: (p_1=\mathbb{P} (\text {coin in a cell for respin})), the size of the multipliers of coins, the chances of special characters (collector, enlarger, double).

For wheel: segment table (probability, prize).

💡 If there are no tables, get them empirically: 2-10 thousand runs of demos/logs, group the outcomes into baskets and estimate the frequencies.

4) How to calculate (\mathbb {P} (X\ge t)) - three practical ways

Method 1: Analytics for Simple Freespins

Let you have (N) frispins, a factor (M), and consider at least one spin with (Y\ge y_0) "significant." Then:

The chance of a "big hit" in one back: (q =\mathbb {P} (Y\ge y_0)).
Chance of not getting a single big hit in a round: ((1-q) ^ N).
So (\mathbb {P} (\text {is ≥}y_0) = 1- (1-q) ^ N).
For a threshold by sum (X\ge t), use a convolution of distributions (or a normal approximation if (N) is large and the tails are moderate).

Method 2: Recursion/Markov for "sticky/ladder"

Define the states (s) (number of wilds, current multiplier, remaining spins). For each state, store:

[
EV (s) =\text {waiting to win from here} ,\quad P_{\ge t} (s) =\text {chance to exceed the threshold}.
]

Calculate from bottom to top: for terminal states, the values are known; for non-terminal:

[
EV(s)=\sum_{s'} p_{s\to s'},[,r(s\to s')+EV(s'),],\quad
P_{\ge t}(s)=\sum_{s'} p_{s\to s'},P_{\ge t'}(s'), ]

where (t ') is the remaining threshold, taking into account the one already dialed.

Method 3: Monte Carlo (universal)

Model the bonus 100k-1M according to their rules. For each, count (X). Then:

(\widehat{EV}=\frac{1}{M}\sum X^{(m)})
(\widehat{\mathbb{P}}(X\ge t)=\frac{#{X^{(m)}\ge t}}{M})
Estimate confidence intervals with bootstrap.
This is the most practical way when the mechanics are complex or the tables are incomplete.

5) Approximate calculations (simplified)

Example A: 10 freespins, multiplier × 2

Let's say the empirical of one spin in a bonus:

(P(Y=0)=0. 60,\ P(Y=0. 5)=0. 25,\ P(Y=2)=0. 10,\ P(Y=10)=0. 04,\ P(Y=50)=0. 01).
Then (\mathbb {E} [Y] = 0\cdot0. 60+0. 5\cdot0. 25+2\cdot0. 10+10\cdot0. 04+50\cdot0. 01=1. 15).
(\Rightarrow \mathbb{E}[X]=N\cdot M\cdot \mathbb{E}[Y]=10\cdot2\cdot1. 15 = 23) bets.
The chance of at least one ≥×10 spin (up to a factor) is (q = 0. 04+0. 01=0. 05).
Chance of getting ≥×10 at least once in 10 spins: (1- (1-0. 05)^{10}\approx 40%).
The chance to exceed in total, say, × 30 - we estimate it in convolution or Monte Carlo.

Example B: Hold & Spin (6 × 3, 3 respins, starting 3 coins)

Let the chance that in the next respin ≥1 new coin will fall, (p = 0. 42). The probability of finishing right now is (1-p = 0. 58).

The expected number of additional respins before the stop (excluding field filling) (\approx\frac {p} {1-p }\approx 0. 72) "continuation cycles."

The probability of filling all 15 cells is small and increases with the presence of expander characters; evaluated by recursion/simulation.

EV - the sum of the average values of coins (taking into account rare upgrades) by the expected number of collected positions.

6) From expectation to risk: spread and quantiles

Heavy tails in bonuses: rare large outcomes form a significant part of EV. Therefore, in addition to EV, consider:

Quantiles (Q_{50},Q_{75},Q_{90}) for (X): what the player "usually" sees;
(\mathbb {P} (X = 0)) or near-zero outcomes (total failure);
(\mathbb {P} (X\ge t)) for multiple thresholds (× 10, × 25, × 50, × 100).
This gives an honest picture: "most often like this," "sometimes - like this," "rarely - like this."

7) Bonus Purchase (Feature Buy)

If the purchase is worth (C) rates, then the net expectation is

[
EV_{\text{net}}=\mathbb{E}[X]-C.
]

If (EV_{\text{net}}<0), then mathematically the purchase is unprofitable, even if it increases the frequency of "action." Compare also the risk profile: buying often increases variance.

8) A "bonus passport" template for your reviews

Bonus type: freespins/hold & spin/wheel/mixed

Parameters: (N), multipliers, special characters, additives, mesh size

Bonus EV: ... rates (method: analytics/Monte Carlo, (M) runs)

Win quantiles (X): (Q_{50}=...), (Q_{75}=...), (Q_{90}=...)

(\mathbb{P}(X\ge ×10 / ×25 / ×50 / ×100)): … / … / … / …

(\mathbb {P} (fail)):...

Risk comment: variance (low/medium/high), typical deserts

Feature Buy: price (C), (EV_{\text{net}}) =...; conclusion on feasibility

9) Frequent errors in assessments

Ignore the state dependency (sticky mechanics) and count as independent spins.

Rely only on the average. Show quantiles and threshold odds.

Mix game versions (different RTP pools) in the same statistics.

Monte Carlo Short Sample for Heavy Tails: Increase runs to 100k +.

10) Short algorithm of actions

1. Write down the bonus rules (steps/states where randomness).

2. Collect/estimate probabilities (tables or empirics).

3. Choose a method: analytics (when simple), recursion (when there are states), Monte Carlo (always works).

4. Calculate EV and (\mathbb {P} (X\ge t)) for multiple (t).

5. Give quantiles and risk inference; when buying - compare with the price.

Bottom line: the chances of winning a bonus count - whether it's freespins, a wheel or hold & spin. The key is to correctly describe the mechanics, choose a suitable model and estimate not only the average (EV), but also the chances of exceeding important thresholds, along with the spread. So you get a realistic picture of risk and expectations, and not the illusion of "timing" or "magic" patterns.