One of the most common questions in probability, gambling, and gaming is: “If something has a 1% chance of happening, and I try 100 times, am I guaranteed to succeed?”
The intuitive answer is “Yes,” but the mathematical answer is “No, you only have a 63.2% chance.”
This is known as the Cumulative Probability of “At Least One Success.” Whether you are a slot player chasing a bonus round, a gamer farming a rare drop, or a bettor looking for one win in a series of longshots, our calculator reveals the reality of variance and helps you plan your attempts accurately.
Chance After N Attempts
BinomialHow to Use the Calculator
This tool uses the complement formula P = 1 − (1 − p)n to calculate your chances. Here is how to configure it:
- Choose Your Input Mode:
- Odds (1 in X): Best for slots, loot drops, or any “1 in X chance” scenario (e.g., “1 in 100”).
- Percentage (%): Best for sports betting or general statistics (e.g., “5%” win probability).
- Enter Single Event Probability: Input the chance of success for a single trial.
- Enter Number of Attempts (N): How many spins, rolls, kills, or bets are you planning?
- Analyze the Results:
- Success probability: Your chance of hitting at least one success in N attempts.
- Failure probability: The chance of getting zero successes (all fails).
- Certainty Milestones: Exactly how many tries you need to reach 50%, 90%, or 99% confidence.
Related tools: if you are analyzing a sequence of consecutive wins or losses, use our Streak Calculator. To check if your slot session results are statistically normal, see the Slot RTP Simulator. For casino bankroll planning, try the Roulette Bankroll Calculator.
The “63% Rule” Explained
Why is 100 attempts at 1% not a guarantee? Because with every failed attempt, the total probability of “all failures” compounds — and it compounds more slowly than people expect.
The formula: P(at least one success) = 1 − (1 − p)n
When the number of attempts equals 1/p (for example, 100 attempts at 1% probability), the result converges to approximately 63.2%. This is the so-called “63% rule” — and it means roughly 1 in 3 people will complete the “expected” number of attempts and get zero successes.
Real-World Examples
Example 1: The Slot Machine Bonus
You are playing a slot where the Bonus Round triggers, on average, once every 100 spins (1% probability).
- You play 100 spins.
- Intuition says: 100% chance to hit.
- Math says: 63.4% chance to hit.
- Reality: Roughly 1 in 3 players will play 100 spins and get zero bonuses.
- To be 99% confident: you need approximately 459 spins.
Example 2: The Rare Loot Drop (Gaming)
You are farming an item in an MMO with a drop rate of 1 in 1,000 (0.1%). You decide to kill the boss 2,000 times.
- After 2,000 attempts: your chance of getting at least one drop is 86.5%.
- After 1,000 attempts (1× the rate): only 63.2% — the 63% rule again.
- Verdict: Even with double the “expected” attempts, there is still a 13.5% chance you walk away empty-handed.
Example 3: Roulette — Hitting Your Number
You want to see the number “7” appear on a European Roulette wheel (1 in 37 chance). You spin 37 times.
- Result: You have a 63.7% chance of seeing the number 7 at least once.
- For 99% confidence: you would need approximately 168 spins.
Example 4: Longshot Betting
You bet on a +500 underdog (implied 16.7% win probability) five times in a row.
- At least one win: 59.7% chance.
- All five lose: 40.3% chance.
- Insight: Even at nearly 60% cumulative probability, more than 2 in 5 bettors will go 0-for-5 on this play. The cumulative math feels more optimistic than the reality.
Frequently Asked Questions (FAQ)
What is the formula for “At Least One Success”?
Calculate the probability of failing every single time, then subtract from 1. The formula is: P(Success) = 1 − (1 − p)n, where p is the probability of a single event, and n is the number of attempts.
Does the machine know I played N times?
No. This calculator assumes independent events (like slots or dice). The machine has no memory. If you missed 99 times, the chance of hitting on the 100th time is exactly the same as the first time. The calculator shows the cumulative probability of the entire session, not the specific odds of the next spin.
Why does the probability never reach 100%?
Mathematically, you can never be 100% certain of a random event outcome in a finite number of trials. Even if you try a million times, there is a microscopically small chance of failing every time. However, you can reach 99.999% certainty, which is a practical guarantee for all real-world purposes.
What is the “63% Rule”?
When the number of attempts equals 1/p (for example, 100 attempts at a 1% chance), the probability of at least one success is always approximately 63.2%, regardless of what p is. This comes from the mathematical limit: as n approaches 1/p, the expression (1 − 1/n)n approaches 1/e ≈ 0.368, so the success probability approaches 1 − 1/e ≈ 0.632.
How many attempts do I need to be 99% sure?
Use the formula: n = ln(1 − 0.99) / ln(1 − p). For a 1% event, that is approximately 459 attempts. For a 0.1% event, approximately 4,603 attempts. The Certainty Milestones section of the calculator shows this automatically for 50%, 90%, and 99% confidence.
Does this work if the odds change between attempts?
No. This calculator assumes a constant per-attempt probability and independent trials. If the probability changes between attempts — for example, pity systems in games, dynamic loot tables, progressive slot mechanics, or drawing without replacement — the formula does not apply directly. For those scenarios, you would need a model that accounts for the changing probability at each step.
Does this calculator work for sports betting?
Yes, with one caveat: you need to express the win probability as a percentage. For example, a +300 underdog has an implied probability of 25%. Enter 25% as the single event probability and the number of bets as N. The calculator will show your chance of hitting at least one winner in the series.