The at-least-one trap
People multiply or add when they should use 1 − (1 − p)ⁿ. A 10% chance over 10 tries is 65%, not 100%.
Tools / Probability Calculator
Probability Calculator
Three modes in one page: simple favorable-over-total probability, the chance of at least one success over several trials, and normal distribution areas.
P = favorable ÷ total · P(at least one) = 1 − (1 − p)ⁿ
Enter values to see the result
- 1
Pick a mode
Simple, at least one, or normal distribution.
- 2
Enter the inputs
Favourable and total outcomes; or a single-trial probability and a number of trials; or a mean, standard deviation, and bounds.
- 3
Read the probability
Shown as a decimal and a percentage, with the complement so you can see both sides.
Why three modes
Most everyday probability questions are one of these three. Keeping them together saves hunting for the right page.
Normal areas without a table
Below, above, or between two bounds, computed from the mean and standard deviation directly.
Complement always shown
The probability of the event not happening comes free, which is often the number you actually wanted.
Typical questions
Risk over repeated exposure
The chance that at least one of 200 orders hits a rare failure.
Quality thresholds
The share of items expected to fall outside a tolerance band, given a mean and standard deviation.
Homework and games
Dice, draws, and coin problems in the simple mode.
The three formulas
Simple: P = favorable ÷ total · At least one: P = 1 − (1 − p)ⁿ · Normal: P = Φ((b − μ)/σ) − Φ((a − μ)/σ)
The at-least-one formula assumes independent trials with the same probability each time. The normal mode uses the standard normal CDF, so it assumes the underlying data is approximately normal.
Common mistakes
- Adding probabilities across trials. Two 50% chances do not make 100%.
- Treating dependent events as independent, for example drawing cards without replacement.
- Using the normal mode on strongly skewed data, where the tail areas will be wrong.
- Confusing the probability of an event with the probability of at least one event across many tries.
Related calculators
Frequently asked questions
How do you calculate simple probability?
Divide favorable outcomes by total possible outcomes:
P = favorable ÷ total
Rolling a 4 or higher on a six-sided die is 3 ÷ 6 = 0.5, or 50%.
How do you calculate the probability of at least one success?
Work out the chance of it never happening and subtract from 1:
P(at least one) = 1 − (1 − p)ⁿ
With a 10% chance per attempt over 10 attempts, that is 1 − 0.9¹⁰ ≈ 0.651, or about 65%. Adding 10% ten times to reach 100% is the classic error.
What does the normal mode calculate?
The area under a normal curve with your mean and standard deviation: below a value, above a value, or between two bounds.
For a mean of 100 and standard deviation of 15, the probability of a value below 130 is about 0.977.
What is the complement of a probability?
The chance the event does not happen, which is 1 − P. It is often much easier to compute, which is exactly why the at-least-one formula works the way it does.
Does this handle dependent events?
No. The at-least-one mode assumes each trial is independent and carries the same probability. Drawing without replacement changes the odds each time, so it needs different maths.
When should I not use the normal mode?
When the data is strongly skewed or bounded, such as income, wait times, or counts near zero. The tail probabilities will be wrong, sometimes badly.
Check the shape with the mean median mode calculator: a mean far from the median is a warning sign.
Are my numbers stored?
No. Everything runs locally in your browser.
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