Formula
P = favourable ÷ totalIndependent: P(A and B) = P(A)·P(B); P(A or B) = P(A) + P(B) − P(A and B)Binomial: P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ
- Mutually exclusive: P(A and B) = 0 and P(A or B) = P(A) + P(B)
Probability runs from 0 (impossible) to 1 (certain). It is the share of equally likely outcomes that count as success.
Worked examples
Rolling a 6
- Situation
- One event (favourable ÷ total outcomes)
- Favourable outcomes
- 1
- Total equally likely outcomes
- 6
Probability
0.166667
As a percentage: 16.6667%
Probability it does not happen: 0.833333
Odds in favour: 1 : 5
Exactly 3 heads in 10 flips
- Situation
- Repeated trials (binomial)
- Number of trials (n)
- 10
- Number of successes (k)
- 3
- Chance of success each trial
- 50 %
P(X = k)
0.1171875
P(X ≤ k): 0.171875
P(X ≥ k): 0.9453125
Expected successes (n × p): 5
How it works
Rolling a fair die and wanting a 6 gives 1 favourable outcome out of 6, so the probability is 1/6 ≈ 0.1667 (16.67%), with odds of 1 : 5 in favour. For two independent events at 50% each, both happen 25% of the time and at least one happens 75% of the time.
The binomial option answers questions like “what's the chance of exactly 3 heads in 10 fair coin flips?” (11.72%) and also gives the chance of at most and at least that many. The number of ways to pick which trials succeed is the combination count C(n, k).
Probabilities here assume the inputs are exact. For real data, remember a small sample can be far from the long-run chance.
Assumptions and limits
- Outcomes in the single-event mode are assumed equally likely.
- Binomial trials are assumed independent with the same chance each time; n is limited to 1,000.