Winning frequency: how probabilities work

1) Basic terms (short)

RNG - server random number generator, sets the outcome of each round.

Hit rate (HR) - frequency of "hits." Clarify what they consider a "hit":

HR\_ any: proportion of rounds with any payout'M> 0'.
HR\_ net: share of rounds with net profit 'M> 1' (payout greater than bet).
RTP - Expected Long Term Return'RTP = Σ p_i·M_i'.
Volatility - the variance of payments (how ragged the profile is).
Max exposure - ceiling of winnings per round (cap multiplier).
Here 'p _ i' is the probability of outcome 'i', 'M _ i' is the corresponding multiplier to the bet.

2) Why high HR doesn't mean plus

RTP is distributed across all outcomes. You can make frequent small payments (high HR\_ any) with low RTP, if the lion's share of "hits" are small coefficients (or returns ≤ 1 ×). In contrast, low HR at rare large factors can yield the same RTP.

Total: HR describes "how often something happens," RTP - "how much is returned on average," volatility - "how much the result jumps."

3) Formulas for one round and for a session

Expected payout per round: 'E [Payout] = RTP· S', where 'S' is the bet.

Expected total for N rounds (with fixed S): 'E [Net] = N· S· (RTP − 1)'.

Probability of ≥ one "hit" in N rounds:

по HR\_any: `P(≥1) = 1 − (1 − HR_any)^N`;
по HR\_net: `P(≥1) = 1 − (1 − HR_net)^N`.
Probability of k "hits" (Bernoulli, independent rounds): 'C (N, k)· HR ^ k· (1 − HR) ^ (N − k)'.
The average number of rounds before the first "hit": '1/HR'.
Before the first bonus with chance p: geometric expectation '1/p'.

💡 In licensed RNG games, rounds are simulated independently. "Series" and "stripes" are normal statistics, not "twisting."

4) Clear example (discrete model)

Let the Tap & Win outcome map be as follows:

Outcome	Probability	'M'multiplier
-------------	----------:	------------:
It is empty	0. 52	0×
Small gain	0. 36	1. 5×
Average	0. 10	3×
Large	0. 02	5×

Checks and conclusions:

RTP: `0. 36·1. 5 + 0. 10·3 + 0. 02·5 = 0. 54 + 0. 30 + 0. 10 = 0. 94` → 94%.
HR\_any = HR\_net = 0. 36+0. 10+0. 02 = 0. 48 (48%) (all payments> 1 ×).
Probability of seeing ≥1 payout for 10 rounds: '1 − 0. 52^10 ≈ 99. 86%`.
Expected result for N rounds with bet S: 'E [Net] = N· S· (0. 94 − 1) = −0. 06·N·S`.

💡 The same RTP = 94% can be collected with low HR (rare large factors). That is why compare HR and RTP together, plus see volatility (spread).

5) Frequencies of "bonuses" and "big winnings"

Developers often publish the frequency of rare events (for example, "bonus falls 1 out of 100" → 'p = 0. 01`).

Chance not to see a bonus for 200 rounds: '(1 − 0. 01)^{200} ≈ 13–14%`. This is normal and not a sign of "dishonesty."

Average interval between bonuses: '1/p' rounds (the law of large numbers works over a long distance, not in a short session).

6) Crash and cashout threshold (general, no "magic")

In crash subspecies, the 'X' multiplier has a survival function 'S (x) = P (X≥x)' specified by the provider.

The chance to "catch" the cache out on the threshold 'x' is' S (x) '.

RTP hardwired to distribution'X '; carrying the threshold changes the variance rather than the expectation of the system.

Practice: early 'x' → above HR\_ net, below large tails; late 'x' → vice versa.

7) How providers collect the right profile

Table of weights/sectors (discrete PMF) or continuous distribution (for crash/physics).

Tuns: frequencies of small payments (manage HR), shares of medium/large (manage RTP and tails), caps cut max exposure.

The independence of the rounds is maintained; progressives/quests should not change the chances of outcome (honest design).

8) How to read "frequencies" correctly

1. Find out what exactly is considered a "hit": any payment or profitable payment.

2. See RTP and mouthguards along with HR. Frequent "noise" at low factors can produce high HR\_ any with low RTP.

3. For rare events (bonus/large X), think in session probabilities rather than "should have fallen."

4. In crash, use auto-cashout (for example, X1. 5-X2) to stabilize HR\_ net and dispersion.

5. Remember: 'E [Net] = N· S· (RTP − 1)' - the pace of the game (N/h) directly affects the hourly exposure.

9) Evaluate HR on your data

Rating: 'HR̂ = k/N' (k is the number of "hits" in N rounds).

Coarse 95% interval: 'HR̂ ± 1. 96·√ (HR̂ (1−HR̂ )/N) '(for large N).

Compare HR\_ net and HR\_ any: the discrepancy shows the proportion of "returns/micro-payments."

10) Frequent misconceptions

"It should alternate: there were a lot of empty ones - now it will fall →" the player's mistake (gambler's fallacy). Independent rounds do not "remember" the past.

"I will catch the timing - I will change the chance" (in instant modes) → the outcome is set by RNG; timing affects only where skill windows are provided.

"High HR = profitable game" → without RTP/volatility the thesis is meaningless.

"Bonus 1/100, so it will definitely fall out for 100" → not, mathematically this is an expectation, not a guarantee.

11) Pre-game checklist (about probabilities)

Is it determined what a "hit" is in this game?

Are RTP, volatility/payout profile, multiplier cap visible?

Is there data on the frequencies of the bonus and large X (threshold by agreement, e.g. ≥ X10)?

Crash: Are auto-cashout and 'S (x)' stats/cache-out history available?

Do you understand your hour exposure 'N· S' and are you planning limits?

12) Responsible play (minimum)

Time/deposit limits, pauses, demos for getting to know HR/RTP/performance. Play with licensed operators (RNG audit); Remember that expectation <1 is the norm for gambling, and HR describes the frequency of events, but not profit.

Result

The win rate in Tap & Win is about how often you see payments, not how much you end up getting. Look at HR\_ any/HR\_ net along with RTP and volatility, consider the chances of a session as' 1 − (1 − HR) ^ N'formulas, use auto-cashout where appropriate, and control the tempo (' N ') limits. Then "frequencies" will become a useful tool of choice and expectation, and not a source of illusions.