A game says your chance is 1 in 100. You play it a hundred times. Surely it has to land by then? It does not. The chance of seeing at least one hit in a hundred tries is about 63%, which leaves better than a one-in-three chance of a hundred misses in a row. The gap between "1 in 100" and "once per hundred tries" trips up a lot of players, and the arithmetic that closes it fits on one line.
The formula: count the misses instead
Working out "at least one hit" directly is messy, because the hit could come on the first try, the fifth, or several times over. The easy route is to calculate the opposite, a run of nothing but misses, and subtract it from one.
- A single miss at 1-in-100 odds has a chance of 0.99.
- Two misses in a row: 0.99 × 0.99. A hundred misses in a row: 0.99 multiplied by itself a hundred times, which is about 0.366.
- So the chance of at least one hit in a hundred tries is 1 − 0.366, about 0.634, or 63.4%.
In general, with a chance of p on each try and n tries, the chance of at least one hit is one minus (1 − p) raised to the power n, often written 1 − (1 − p)^n. The only condition is that tries are independent: each spin, draw or deal starts fresh, with no memory of what came before.
How a 1-in-100 chance builds up
| Tries | Chance of at least one hit |
|---|---|
| 1 | 1% |
| 10 | 9.6% |
| 50 | 39.5% |
| 69 | 50.0% |
| 100 | 63.4% |
| 200 | 86.6% |
| 300 | 95.1% |
| 459 | 99.0% |
Three things stand out. The chance climbs quickly at first and then slows, because each extra try can only rescue the shrinking share of runs that are still all misses. Even at 459 tries there is a 1% chance of no hit at all. And no number of tries ever makes it 100%.
Mean wait and median wait
For a 1-in-100 event the average wait for the first hit is exactly 100 tries. Yet half of all waits are over by the 69th try. Both are true at once because the waits are lopsided: most are fairly short, but a few run very long, two or three hundred tries, and those long tails drag the average upwards.
A handy shortcut for the median is to multiply the "N" in 1-in-N by about 0.69. A 1-in-37 shot, the chance of a single number on a single-zero roulette wheel, has a median wait of 26 spins, and only reaches a 63.7% chance of having appeared after 37 spins. The same shape holds for very long odds: for a five-card royal flush at 1 in 649,740, even 649,740 hands give only about a 63% chance of at least one. The long-shot calculator on our page about the odds of a royal flush runs these numbers for any odds you type in.
Where players go wrong with it
- "It is due." A long run of misses does not raise the chance on the next try. With independent tries the 1-in-100 chance is the same on the 300th attempt as on the first; the formula above is about whole runs, looked at from the start.
- "A hundred tries will cover it." As the table shows, a hundred tries at 1 in 100 still miss completely more than a third of the time. Budgeting for "one full cycle" of the odds is not budgeting for a hit.
- Adding chances together. Ten tries at 1% each do not give 10%; they give 9.6%. The error is small for a few tries but grows badly: adding would claim 150 tries gives 150%, when the true figure is about 78%.
- Forgetting the price. At-least-one maths says nothing about whether a hit pays for the tries it took. A slot feature or a long-shot bet can land and still leave the session behind, because the house edge is built into the payout. How those payouts are set is described in our explainer on slot machine PAR sheets.
Using the numbers sensibly
The most practical use of this formula is setting a limit before a session rather than chasing a result during it. If a target outcome has, say, a 40% chance of appearing within the number of spins your budget covers, then a 60% chance of it not appearing is part of the plan, not bad luck. Number games are a clear example: a single pick from a hundred possibilities follows exactly the table above, and the guide to responsible play in number games like Satta King sets out the same idea in practical terms. For slot sessions kept small on purpose, the piece on enjoying online slots on a low budget is a good companion.
Games of chance are for adults only. Decide your spend before you begin, stop when it is used up, and check the rules where you live, because the legality of online play differs from country to country.