What Is Probability 1 Event?
The probability of one event is its favourable outcomes divided by the total. When you repeat the event, the chance of seeing it at least once grows, even though each trial has the same probability.
Enter favourable and total outcomes, or the probability itself, then the number of trials and an exact count k. You get the single-trial probability, the chance of at least one, none and exactly k, the expected count and the number of trials for a 50 and 95 percent chance.
Below are the formulas, a worked dice example and answers on independence.
How to Use the Probability 1 Event Calculator
- Enter favourable and total outcomes, or p.
- Enter the number of repeats.
- Enter k for an exact count.
- Read each probability and the trials needed.
Probability 1 Event Formula
At least once
Exactly k
Probability 1 Event Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| p | Probability on one trial | |
| n | Number of trials | |
| k | Exact number of occurrences | |
| 1 − p | Probability the event does not happen |
Probability 1 Event Worked Example
Rolling a six has p = 1/6 = 0.1667. In 10 rolls the chance of no six is (5/6)^10 = 0.1615, so the chance of at least one six is 0.8385. About 4 rolls give a 50 % chance and 17 rolls a 95 % chance.
With the example values loaded in the calculator above, the steps are:
- Single trial: P(event) = 0.16666667.
- P(never in 10) = (1 − p)^10 = 0.16150558.
- P(at least once) = 1 − 0.16150558 = 0.83849442.
- P(exactly 2) = C(10, 2)·p^2·(1 − p)^8 = 0.29071005.
Result: P(event) on one trial = 0.16666667; P(at least once in 10 trials) = 0.83849442; P(exactly 2 times in 10 trials) = 0.29071005.
How to Read a Single-Event Probability
The probability p of one try is just the starting point. Repeating the try changes the picture: the chance of at least one success in n tries is 1 − (1 − p)ⁿ, which climbs toward 1 but never reaches it.
For a six on a fair die, p = 1/6 ≈ 0.167. In 10 rolls the chance of at least one six is 1 − (5/6)¹⁰ ≈ 0.839. It takes about 4 rolls to pass 50% and 17 rolls to pass 95%.
Where Probability 1 Event Is Used
Single trials
Convert a fraction, decimal or percentage into the others, and get the odds in favour and against.
Repeated attempts
Find the chance of at least one success, or exactly k successes, over n independent tries.
Planning
Work out how many tries you need to reach a 90%, 95% or 99% chance of at least one success.
Risk talk
Express a small risk such as 1 in 500 per year as the cumulative chance over 10 or 20 years.
Common Probability 1 Event Mistakes to Avoid
- The tries must be independent and have the same p each time. If p changes, the formulas do not apply.
- "At least one" is easiest as 1 minus "none". Do not add the single-try probabilities.
- A 50% chance per try does not mean 2 tries guarantee a success. It means 75%.
- Use the expected count n·p for the average, and the probability for the chance of a particular outcome.
Probability 1 Event Terms and Definitions
- Complement
- The event not happening, with probability 1 − p.
Probability 1 Event FAQs
How do you find the probability of a single event?
Divide the number of favourable outcomes by the number of equally likely outcomes. For a six on a fair die that is 1 ÷ 6 ≈ 0.1667, or 16.7%.
How do I find the chance of at least one in n trials?
Compute 1 − (1 − p)ⁿ. With p = 1/6 and n = 10 the chance of no six is (5/6)¹⁰ ≈ 0.1615, so at least one six has probability 1 − 0.1615 = 0.8385.
How many trials to reach a 95 % chance?
Solve 1 − (1 − p)ⁿ ≥ 0.95, which gives n ≥ ln(0.05) ÷ ln(1 − p). For p = 1/6 that is about 16.4, so you need 17 rolls.
What is the expected number of occurrences?
It is n × p. In 60 rolls of a die you expect about 10 sixes on average, but any single run of 60 rolls will usually differ from 10.