What Is Probability and Chance?
The probability of an event is a number from 0 to 1 (or 0 % to 100 %) that describes how likely it is. Odds describe the same thing as a ratio, such as 1 to 3, and "1 in N" is a way of saying it in words.
Repeated independent tries change the picture. The chance of at least one success in n tries is 1 − (1 − p)ⁿ, and the chance of exactly k successes comes from the binomial formula. The calculator gives both, the expected number of occurrences and the number of tries needed to reach a chosen chance.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as the difference between probability and odds.
How to Use the Probability and Chance Calculator
- Enter the event either as favourable over total outcomes or directly as a probability.
- Enter how many times you will try and the number of successes k you want to look at.
- Read the single-event chance, odds and 1-in-N, then the repeated-trial results.
- Use the “tries needed” rows to plan how many attempts give you a good chance of success.
Probability and Chance Formula
Classical probability
Odds in favour
At least once in n independent trials
Exactly k successes in n trials
Tries needed for a target chance q
Probability and Chance Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| p | Probability of the event in one try | 0 to 1 |
| n | Number of independent tries | |
| k | Number of successes | For the "exactly k" result |
| Odds | p ÷ (1 − p) | Odds in favour |
Probability and Chance Worked Example
Using the example values (press “Load example” above to try them):
- Single-trial probability p = 1 ÷ 6 = 0.16666667.
- At least once in 10 trials = 1 − (1 − p)ⁿ = 1 − 0.833333^10 = 0.83849442.
- Exactly 2 in 10: C(10, 2) × p^2 × (1 − p)^8 = 45 × 0.166667^2 × 0.833333^8 = 0.29071005.
Result: Probability of the event (single trial) = 0.16666667; At least once in 10 trials = 0.83849442; Exactly 2 times in 10 trials = 0.29071005.
Probability versus odds
People mix these up all the time. Probability compares the event to everything that could happen; odds compare it to the alternative. A probability of 1/6 is 1 to 5 odds in favour (one way to win to five ways not to), and 5 to 1 odds against.
| Probability | Percentage | Odds in favour | Odds against |
|---|---|---|---|
| 1/2 | 50% | 1 to 1 | 1 to 1 |
| 1/4 | 25% | 1 to 3 | 3 to 1 |
| 1/6 | 16.67% | 1 to 5 | 5 to 1 |
| 1/10 | 10% | 1 to 9 | 9 to 1 |
| 1/100 | 1% | 1 to 99 | 99 to 1 |
The surprising maths of repeated tries
A 1-in-6 chance does not become certain after six tries. The chance of at least one success in six tries is 1 − (5/6)⁶ ≈ 66.5%. In general, if the chance per try is p, you need about 0.69 ÷ p tries for even odds of at least one success, and about 3 ÷ p for 95%. This is why rare risks accumulate: a 1% daily risk gives about a 63% chance of at least one occurrence over 100 days.
| Chance per try | Tries for 50% overall | Tries for 95% overall |
|---|---|---|
| 50% | 1 | 5 |
| 25% | 3 | 11 |
| 10% | 7 | 29 |
| 5% | 14 | 59 |
| 1% | 69 | 299 |
| 0.1% | 693 | 2,995 |
Assumptions
- Each try is independent of the others.
- The probability stays the same on every try (no learning, wear or “being due”).
- Outcomes are either success or failure.
- The gambler’s fallacy: a past run of failures does not make success more likely next time.
Where Probability and Chance Is Used
Converting probability and odds
Switching between percentages, odds and "1 in N".
Chance over several attempts
Finding how likely at least one success is after n independent tries.
Planning attempts
Finding how many tries give a chosen chance of success.
Classroom problems
Checking answers on classical and repeated-trial probability.
Common Probability and Chance Mistakes to Avoid
- “At least one” is almost always best solved as 1 minus the chance of none.
- Check that probabilities of exclusive, exhaustive outcomes add to 1.
- Use 1-in-N figures with care: 1 in 100 is not the same as “once in 100 tries”.
Probability and Chance Terms and Definitions
- Odds
- The ratio of the chance an event happens to the chance it does not.
- Trial
- One attempt or repetition of the experiment.
Probability and Chance FAQs
How do you calculate probability?
Divide the number of favourable outcomes by the total number of equally likely outcomes.
How do I calculate the chance of at least one success in several tries?
Compute 1 − (1 − p)ⁿ, where p is the chance per try and n is the number of tries.
What is the difference between odds and probability?
Probability is favourable over total. Odds are favourable over unfavourable. A probability of 0.2 is odds of 1 to 4.
How many tries do I need for a 50% chance?
About 0.69 divided by the per-try probability. The calculator gives the exact number.
If the chance is 1 in 6, will I succeed in 6 tries?
Not necessarily. The chance of at least one success in six tries is about 66.5%, not 100%.
Can I enter percentages?
Yes, switch to the probability input and enter a decimal, a percentage such as 16.7% or a fraction such as 1/6.