How to calculate the chance of winning over a long distance

1) What exactly do we think

We are interested in the probability of being in the black after (N) attempts at given payout rules and the chance of winning one attempt. The model is different for different games:

Bets 1:1 (roulette, even/odd, red/black): discrete binomial model.
Slots: payments are of different sizes, the normal approximation in mean and variance is more convenient.

The main idea: with EV <0 (edge> 0), the chance of "being in positive territory" decreases with the growth of N. With EV> 0, it increases, but depends on the variance.

2) Term base

RTP - average return (in fractions), edge = 1 − RTP.

EV of one attempt (in 1:1 bids paying 1:1): (EV = p\cdot (+ 1) + (1-p )\cdot (-1) = 2p-1).

Sales volume (=) bid × number of attempts.

The law of large numbers: the average result stretches to (EV) for large (N).

3) 1: 1 stakes: exact formula via binomial distribution

Let (p) be the probability of winning one bet, (q = 1-p), bet = 1 unit, payout 1:1. For (N) bets, the number of wins (W\sim\text {Bin} (N, p)).

Total (S = (+ 1 )\cdot W + (-1 )\cdot (N-W) = 2W - N).

The plus condition is (S> 0\iff W> N/2). Then

[
\boxed{;\Pr(S>0)=\sum_{w=\lfloor N/2\rfloor+1}^{N}\binom{N}{w}p^w q^{N-w};}
]

Example (European roulette, 1:1): (p = 18/37\approx 0. 4865), (q\approx0. 5135).

(N = 50): count the tail of the binomial distribution (W> 25).

(N = 500): condition (W> 250). The tail becomes significantly smaller due to (p <0. 5).

Normal approximation (rapid estimate): at large (N), [

W \approx \mathcal{N}(Np,;Np q),\quad

\Pr(S>0)\approx 1-\Phi!\left(\frac{N/2 - Np}{\sqrt{Np q}}\right), ]

where (\Phi) is the EME of normal law.

4) Bets with different payout (e.g. (k!: ! 1))

If you pay (k) units for winning with probability (p), and losing is 1 unit, the result is:

[
S = kW - (N-W) = (k+1)W - N.
]

The plus condition is (W >\dfrac {N} {k + 1}). Then

[
\Pr(S>0)=\sum_{w=\lfloor N/(k+1)\rfloor+1}^{N}\binom{N}{w}p^w(1-p)^{N-w}.
]

Fast EV check: (EV = kp - (1-p) = (k + 1) p-1). If (EV <0), the plus chance drops with growth (N).

5) Slots: normal approximation by mean and variance

In slots, the payout of one attempt (X) has expectation (\mu = RTP - 1 = -edge) (in fractions of a bet) and variance (\sigma ^ 2) (slot/volatility dependent). Amount per (N) spin:

[
S_N \approx \mathcal{N}\big(N\mu,; N\sigma^2\big).
]

Plus chance:

[
\boxed{;\Pr(S_N>0) \approx 1 - \Phi!\left(\frac{0 - N\mu}{\sigma \sqrt{N}}\right)
= 1 - \Phi!\left(\frac{-N(-edge)}{\sigma \sqrt{N}}\right)
= 1 - \Phi!\left(\frac{edge\sqrt{N}}{\sigma}\right);}
]

Intuition: with a fixed edge> 0, the denominator grows as (\sqrt {N}), so the probability of a plus decreases with increase (N). The higher the (\sigma) (volatility), the slower the waning (wider the tails).

Marks (\sigma) "on fingers":

Average volatility slots: (\sigma) one attempt ≈ 1. 5-3 stakes.
High volatility: ≈ 3-6 bets.
Substitute in the formula to estimate the order of magnitude.

6) Confidence intervals "where will I be" after N

Via CPT:

[
S_N \approx N\cdot EV \pm z_{\alpha}\cdot \sigma\sqrt{N}.
]

For 1:1 roulette, take a (\sigma _ {\text {one} }\approx 1) bet.

For slots, use the landmarks (\sigma) above.

This gives a "corridor" where the result is likely to fall. If "0" lies far to the right of the mean (N\cdot EV) at EV <0, the plus chance is small.

7) Fast mini calculators

A. 1: 1 tape measure (normal approximation)

[
z=\frac{N/2 - Np}{\sqrt{Np(1-p)}},\quad \Pr(\text{плюс})\approx 1-\Phi(z).
]

B. Overall case k: 1

[
z=\frac{N/(k+1) - Np}{\sqrt{Np(1-p)}},\quad \Pr(\text{плюс})\approx 1-\Phi(z).
]

C. Slots

[
\Pr(\text{плюс})\approx 1-\Phi!\left(\frac{edge\sqrt{N}}{\sigma}\right), \quad \text{где } edge=1-RTP.
]

8) Specific examples

Example 1 - 1:1 tape measure, (N = 200).

(p=18/37\approx0. 4865), (Np=97. 3), threshold (N/2 = 100).

(\sigma=\sqrt{Np(1-p)}\approx \sqrt{200\cdot0. 4865\cdot0. 5135}\approx 7. 07).

(z=(100-97. 3)/7. 07\approx 0. 38) → (\Pr (\text {plus} )\approx 1-\Phi (0. 38)\approx 35%).

Example 2 - 1:1 tape measure, (N = 1000).

(Np=486. 5), threshold 500, (\sigma\approach 15. 8), (z\approx 0. 85) → (\Pr (\text {plus} )\approach 19. 7%).

Growth (N) reduces the chance of plus (EV <0).

Example 3 - RTP slot 96%, average volatility.

edge=0. 04, let (\sigma) one attempt = 2 bets.

(N=1000): (\dfrac{edge\sqrt{N}}{\sigma}=\dfrac{0. 04\cdot31. 62}{2}\approx 0. 632) → (\Pr (\text {plus} )\approx 1-\Phi (0. 632)\approx 26. 4%).

(N = 10,000): measure (\approach 2. 0) → (\Pr (\text {plus} )\approach 2. 3%).

9) How to use calculations in practice

Know the frames: with EV <0, the long distance works against you - the chance of a plus decreases.

Under the goal - the volatility profile: for tournaments/teik-profits with asymmetry, you can prefer high-vol (more tails), but with a lower share of the bet.

Rate in% of bankroll (BR):

high-vol: 0. 25–0. 75% BR, medium: ~ 1% BR, low/1: 1: 1-2% BR.
Play in series: limit (N) in session - control the probability of "going minus far."
Speed control: "price per hour" (\approach edge\times\text {bid }\times\text {attempts/min }\times 60).
Vager: Cost (\approach\text {Bonus }\times\text {Vager }\times edge). Over a long distance, the result gravitates towards this price.

10) Frequent interpretation errors

"After a series of minuses, the chance of a plus grows." No: independence of outcomes.

"I will increase the rate - I will increase the chance of a plus at a distance." No: you will increase turnover and variance, not (p) and not RTP.

"If you hold out long enough, I'll come out as a plus." With EV <0, the probability of the opposite is higher.

11) Checklist (in 60 seconds)

1. Know (p), (k) (or RTP/edge and order (\sigma))?

2. Calculated the winning threshold: (N/2) (1:1) or (N/( k + 1))?

3. Estimated (\Pr (\text {plus})) by the tail of the binomial or by the normal z?

4. Rate set to% of current BR?

5. Is there a limit (N) per session and stop levels (SL/TP)?

6. Speed/" price of the hour" under control?

The chance of "being in positive territory" after (N) attempts is determined by expectation and spread: at EV <0 it decreases with increasing distance (especially in equilibrium rates 1:1), at EV> 0 it increases, but the pace depends on volatility. Use binomial tails for simple bets and normal approximations for slots, keep the bet at% of bankroll, play in series and control speed - this way you turn abstract theory into understandable decisions about the risk and duration of the game.