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Winning math: What every player needs to know
Why does a player need mathematics
Games happen, but math doesn't. Casinos and bookmakers rely on clear numbers: probability, advantage (house edge), variance, the law of large numbers. If you understand these ideas, you choose your bets consciously, run a bankroll and don't fall into "almost won" traps.
1) Basic concepts without which it is impossible
Probability (p). Event chance. 0 to 1 (or 0-100%).
Expected value (EV). Long-distance average. If EV is negative - "entertainment price." If positive, a mathematical advantage.
RTP and casino advantage. RTP slot 96% = 96% average return over long distance; casino advantage = 1 − RTP = 4%. In rates 1 × 2, the advantage is "sewn up" in line margin.
Variance (volatility). How much the result "chatters" around expectation. High variance - rare but large winnings; low - frequent but small.
2) EV on fingers (and numbers)
Coin 50/50
We put 1 unit, win + 1 with eagle, − 1 with tails.
EV = 0. 5·(+1) + 0. 5·(−1) = 0. But there is almost no such bet in casinos.
European roulette - bet on "red"
Probability of winning p = 18/37, losing q = 19/37, payout 1:1.
EV = 1·(18/37) − 1·(19/37) = −1/37 ≈ −2. 7027% of the rate.
Important: the bet on one number (35:1) has the same EV: (35· 1/37 − 36/37) = − 1/37.
Slots
RTP 96% ⇒ average expectation − 4% of turnover. For 100 spins of 1 cu., the expected result ≈ − 4 cu. But the actual result will "jump" due to variance (see below).
3) Variance and why do you need to know about the variance
For even-money rates in roulette (± 1 cu per spin), the variance per 1 spin is close to 1, the standard deviation σ ≈ 1.
N spins σ_N ≈ √N.
Example: for 100 spins, the expected result ≈ − 2. 7 cu and standard deviation ≈ 10 cu. That is, real results often lie in the interval from about − 13 to + 7 cu. (one σ). Therefore, the "short distance" is able to give a plus, although the expectation is negative.
The spread of the slots is even higher: many zero spins and rare major hits. Hence the serial "drawdowns" and sudden ups.
4) The law of large numbers (and why there will be no "long haul")
The larger the turnover, the closer the actual result is to EV. With negative expectations and sufficient game volume, the result will almost certainly go into minus the value of house edge × turnover.
5) Bankroll and the risk of ruin
Bankroll - the amount you are willing to lose without compromising the budget.
Fix limits: the size of one rate (1-2% bankroll for low variance, ≤0. 5-1% for highly dispersed slots), day/session loss limit, time limit.
Risk of Ruin (RoR). With a negative EV, RoR tends to 100% over time: the longer and larger the rates, the higher the chance of "getting to zero." Your task is not to "defeat mathematics," but to control the exposure: lower rates, shorter sessions, clear stop loss.
6) Betting management: When Kelly is useful
Kelly (for positive expectation): f = edge/ odds_variance (in simple bets - approximately f ≈ (bp − q )/b, where b is the net payout coefficient, p is the probability of winning, q is the loss).
Applicable in bets with a real advantage (value-bet in sports, arbitration, rare shares). In classic casino games, there is no preponderance, Kelly should not be used for overclocking, but only as a reminder: without edge, rampant bet growth will only accelerate drawdown.
7) Practice: How to read bonuses and vager
Wager expectation formula:- Expected "tax" on wagering ≈ (Turnover on wagering) × (house edge).
- Turnover = Wager × Bonus Amount (if wagering on bonus only).
Example 1 (rather bad): bonus 100 cu., Vager × 40, RTP games 96% (edge 4%).
Turnover = 100 × 40 = 4000 cu.
Expected "tax" = 4000 × 4% = 160 cu.
Pure Math: + 100 − 160 = − 60 CU (excluding bet/play restrictions).
Example 2 (rarely available): same conditions, but RTP 99. 5% (edge 0. 5%).
Tax = 4000 × 0. 5% = 20 cu.
Pure mathematics: + 100 − 20 = + 80 cu. It is no coincidence that such games are often excluded from the game.
Conclusions on bonuses:- See × edge vager - this is the main "hidden tariff."
- Check game exclusions, bid limits, expiration dates.
- Wager-free cashback is closer to a "net" return, but its percentage tends to be less.
8) Frequent mistakes and myths
"Hot/cold slots. "The backs are independent, RTP is implemented at a distance of. "Stripe" is a manifestation of dispersion, not "recoil tuning."
Doubling to infinity. Martingale doesn't eliminate edge. Table restrictions and final bankroll with negative EV lead to rare but ruinous disruptions.
"A little more will give. "RNG has no memory: the next spin does not "have" to compensate for the past.
Too big bets. Even with neutral EV, a high share of bankroll dramatically increases the risk of rapid drawdown.
9) Formula mini cheat sheet
EV (in casino betting): EV = Σ [win _ i × p_i] − bet.
Casino advantage: edge = 1 − RTP.
Expected result at a distance: Result ≈ Turnover × (− edge).
Standard deviation of a series (roughly): σ _ series ≈ √N × σ _ odnoy_stavka.
"Price" of the vager: Vager × Bonus × edge.
10) How to apply in practice - step by step
1. Before the game: determine the goal (to have fun, not "earn"), budget, time limit.
2. Game Selection: Look for high RTP/low margin; avoid bets with a notoriously large edge (for example, American roulette is worse than European: ≈5. 26% vs. 2. 70%).
3. Bet size: 0. 5-1% bankroll for high dispersion slots, up to 1-2% for low dispersion.
4. Bonuses: count "tax" = vager × edge × bonus; compare with bonus size.
5. During the session: fix stop loss and teik profit (for example, − 20% and + 30% of the session budget); do not raise the bid "to recoup."
6. After: Note the sales volume, total, rates used. This is "your data" for a sober assessment of variance and habits.
Main thing
Math doesn't promise a win today, it explains what you're playing.
The higher the edge and dispersion, the more expensive the entertainment at a distance and the sharper the result.
Managing bankroll and understanding expectation is your way of making the game controlled and safe.