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Mathematics · pattern F, Dataset

Single and combined event probabilities.

Enter favourable and total outcomes for a single probability, then combine two independent events for and, or, and at least once.

Inputs

The formula used

P(A) = favourable ÷ total; P(A and B) = P(A)P(B); P(A or B) = P(A) + P(B) − P(A and B)

The multiplication rule shown assumes independence. The addition rule subtracts the overlap, which is what prevents double counting.

Probability

0.250000

As a percentage
25.0000 %
Odds in favour
13 : 39
Complement, P(not A)
0.750000
P(A and B), independent
0.125000
P(A or B)
0.625000
P(at least one A in the trials)
0.762695

At least once is one minus never.

Computing the probability of at least one success directly means summing many cases; computing the probability of none and subtracting from one takes a single step. That complement trick is the most useful technique in elementary probability, and it is what the last row above does.

Questions about probability

Odds or probability?
Probability is favourable over total; odds are favourable to unfavourable. A probability of 1/4 is odds of 1 : 3.
When can I multiply probabilities?
Only for independent events. If one outcome changes the other's probability, use conditional probability.
What is the gambler's fallacy?
Believing past independent outcomes change future ones. A fair coin after ten heads still has probability one half.

A calculator handles the arithmetic. It cannot teach you the method.

If the number is not the part you are stuck on, that is what the service is for — a specialist who explains the working, not just the answer.

Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.