Big Win Math: How It's Calculated
Introduction
A large gain in the slot is the result of a rare combination of many events: bonus drops, activation of multipliers and symbol cascades. To understand the odds and plan a bankroll correctly, you need to understand the mathematics of payments: RTP, EV, variance and probability distribution. Below are the key concepts and formulas that underlie the calculations of large winnings.
1. RTP and expected value (EV)
Return to Player (RTP) - theoretical percentage of bets returned to the player in the long term:
where $ P _ i $ is the probability of outcome $ i $, $ W _ i $ is the payout multiplier (in units of the bet).
Expected Value for one spin:
where $ S $ is the bet size. EV never guarantees a specific win, but shows the average income for a large number of spins.
2. Variance and standard deviation
The variance $\sigma ^ 2 $ measures the spread of payouts around the EV:
3. Probability distribution of large winnings
Jackpot probability (maximum multiplier $ M $):
In practice, for slots with a declared potential of × 1,000- × 10,000, the "maximum" frequency ranges from $10 ^ {-6} $ to $10 ^ {-8} $.
The laws of large numbers guarantee the approximation of the actual average to RTP at $ No\infty $, but the jackpot event even at $ N = 10 ^ 7 $ spins remains rare.
4. Estimating the number of spins before a big win
Geometric distribution model. If the probability of a large event $ p $, then the average number of spins before the first such event:
5. Confidence interval and stability of results
Standard error of mean (SE) for $ N $ spins:
A large gain in the slot is the result of a rare combination of many events: bonus drops, activation of multipliers and symbol cascades. To understand the odds and plan a bankroll correctly, you need to understand the mathematics of payments: RTP, EV, variance and probability distribution. Below are the key concepts and formulas that underlie the calculations of large winnings.
1. RTP and expected value (EV)
Return to Player (RTP) - theoretical percentage of bets returned to the player in the long term:
- $$
- \mathrm{RTP} = \sum_{i} P_i imes W_i,
- $$
where $ P _ i $ is the probability of outcome $ i $, $ W _ i $ is the payout multiplier (in units of the bet).
Expected Value for one spin:
- $$
- \mathrm{EV} = S imes \frac{\mathrm{RTP}}{100},
- $$
where $ S $ is the bet size. EV never guarantees a specific win, but shows the average income for a large number of spins.
2. Variance and standard deviation
The variance $\sigma ^ 2 $ measures the spread of payouts around the EV:
- $$
- \sigma^2 = \sum_{i} P_i imes (W_i - \mu)^2,
- \quad
- \mu = \frac{\mathrm{EV}}{S}.
- $$
- The standard deviation $\sigma =\sqrt {\sigma ^ 2} $ indicates how much the results deviate from the EV on average.
- Slot volatility is directly related to $\sigma $: the higher the $\sigma $, the more "jump" wins and the longer the periods without payments.
3. Probability distribution of large winnings
Jackpot probability (maximum multiplier $ M $):
- $$
- P_{ext{max} }\approach\frac {ext {bonus frequency}} {ext {number of possible bonus outcomes}}
- $$
In practice, for slots with a declared potential of × 1,000- × 10,000, the "maximum" frequency ranges from $10 ^ {-6} $ to $10 ^ {-8} $.
The laws of large numbers guarantee the approximation of the actual average to RTP at $ No\infty $, but the jackpot event even at $ N = 10 ^ 7 $ spins remains rare.
4. Estimating the number of spins before a big win
Geometric distribution model. If the probability of a large event $ p $, then the average number of spins before the first such event:
- $$
- E[N] = \frac{1}{p}.
- $$
- Example. For $ p = 10 ^ {-6} $, $ E [N] = 1\, 000\, 000 $ spins. At €1 per spin, the bankroll must cover these backs in order to "live" to the jackpot on average.
5. Confidence interval and stability of results
Standard error of mean (SE) for $ N $ spins:
- $$
- \mathrm{SE} = \frac{\sigma}{\sqrt{N}}.
- $$ 95% confidence interval for average winnings:
- $$
- \mathrm{EV} \pm 1{,}96 imes \mathrm{SE}.
- $$
- $$
- f^= \frac{bp - q}{b},
- $$
For highly volatile slots, SE remains large even at $ N = $100\, $000, so short sessions give results very different from EV.
6. Bankroll Planning: Kelly and Fixed Stakes
1. Fixed share rule
- Allocate for one session no more than 1-2% of the total bankroll. This will limit losses during a prolonged "dry" period.
2. Kelly's criterion
- Allows to optimize the bet size $ f ^ * $ by the formula:
where $ b $ is the winning "ratio" (EV/S - 1), $ p $ is the winning probability, $ q = 1 - p $. For slots with low $ p $ and high $ b $, the result is often negative, indicating an aggressive risk.
7. Combination of different strategies
"Marathon" and "hunting"
- Marathon: Low-voltage slots, EV close to RTP, need moderate bankroll.
- Hunting: highly volatile, multimillion-dollar potentials, bankroll for hundreds of thousands of spins.
Split sessions
- Divide the overall plan into series of 5,000-10,000 spins, analyze the results and adjust the rates.
8. Example of calculation for a major event
Let's say a slot with
bet $ S = €1 $,
declared maximum $ M = × 5\, $000,
bonus frequency of 1% and within the jackpot bonus with a probability of 0.01%.
Then $ p = 0. 0001% $ = $10 ^ {-6} $, and
$$
E [N] = 1\, 000\, 000ext {spins} ,\quad
\ text {potential winnings} = €5\, 000,
$$
that is, on average, for €1,000,000 spent euros, you will receive €5,000 once - negative EV for "hunting" requires an additional source of income (RTP-EV).
Conclusion
Large slot wins are the result of an extremely low probability of high multiplier events. To calculate the odds and plan a bankroll, you need to understand RTP, variance, probability distribution and the average number of spins before the jackpot. By applying the EV, SE and expected spin number formulas, as well as fixed share strategies or Kelly's criterion, you can build a sound "bank hunt" approach and minimize financial risks.
