Counting combinations is only the first step to a probability
A count becomes a probability only after you define the event and an appropriate equally likely sample space.
Permutation and combination calculator →Start with an unordered draw
Imagine choosing two distinct numbers from 1 through 6, without replacement and ignoring order. There are C(6,2)=15 equally likely pairs under a uniform draw. One specified pair, such as {2,5}, therefore has probability 1/15, approximately 6.67%. Counting ordered pairs gives 30, but the event then has two matching orders, so 2/30 gives the same probability.
Count favorable outcomes consistently
There are three even numbers, so pairs containing two even numbers number C(3,2)=3. The probability is 3/15=1/5, or 20%. Use the same ordering convention in numerator and denominator. Mixing an ordered numerator with an unordered denominator can produce a misleading fraction, even if each separate combinatorial count was calculated correctly.
Equal likelihood is an assumption
The ratio of favorable to total counts works only when the counted outcomes have equal probabilities. Weighted drawings, duplicate entries or biased mechanisms require a different model. The calculator counts selections; it does not inspect how a real draw is performed or predict a winning outcome. More possible combinations describe the sample space, not evidence that a particular pattern is more likely to appear next. The even-pair example can also be checked by listing {2,4}, {2,6} and {4,6}, rather than relying only on a formula.
Try the calculators
- Permutation and combination calculator
Count ordered and unordered selections of r objects from n distinct objects.
- Random number
Generate a random integer in an inclusive range.
- Scientific calculator
Arithmetic, powers, trigonometry and logarithms.