Probability Basics
8 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
Probability measures how likely something is, on a scale from 0 (impossible) to 1 (certain). It is the mathematics of chance and it underlies games, weather forecasts, medical tests and insurance. This guide starts from first principles.
The basic formula
If all outcomes are equally likely, P(event) = favourable outcomes ÷ total outcomes. The chance of rolling an even number on a die is 3/6 = 1/2. Probabilities can be written as fractions, decimals or percents.
Sample space and a two-dice example
The list of all possible outcomes is the sample space. Two dice have 36 outcomes. Six of them add up to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so P(sum of 7) = 6/36 = 1/6, the most likely total.
Complements and “at least one”
P(not A) = 1 − P(A). The probability of at least one head in three flips is 1 − (1/2)³ = 7/8. Working out the complement (no heads) is often easier.
Independent and dependent events
Independent: one event does not affect the other, so P(A and B) = P(A) × P(B). Two heads in a row is 1/2 × 1/2 = 1/4. Dependent: the first changes the second. The chance of drawing two aces from a deck without replacement is 4/52 × 3/51 = 1/221.
Combine and test these rules with the probability calculator.
Expected value
The average outcome over many trials: multiply each outcome by its probability and add. A fair die has an expected value of (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5.
Common mistakes
- The gambler’s fallacy: after five heads, tails is not “due”. Each flip is independent.
- Confusing probability with odds. A probability of 1/4 is odds of 1 to 3.
- Adding probabilities of overlapping events without subtracting the overlap.
Test yourself with the probability quiz.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- What is the chance of rolling two 4s with two fair dice?
Show answer
P(A and B), independent events: 0.027777778 · 2.77778% · 1/36
- How many ways can 3 medals be awarded among 8 runners?
Show answer
Combinations C(8, 3): 56 · Permutations P(8, 3): 336
- What is 7!?
Show answer
7!: 5,040
Frequently asked questions
What is the range of a probability?
0 to 1, or 0% to 100%.
What is the difference between theoretical and experimental probability?
Theoretical comes from counting equally likely outcomes; experimental comes from the results of trials.
How do I find the probability of A or B?
P(A) + P(B) − P(A and B).
What is conditional probability?
The probability of an event given that another has occurred: P(A | B) = P(A and B) ÷ P(B).
Try it yourself
Probability CalculatorFind the probability of “not A”, “A and B”, “A or B” and conditional probability.Permutation and Combination CalculatorFind nPr, nCr, factorials and counts with repetition.
Factorial CalculatorCompute n! exactly for n up to 1,000, with digit count and trailing zeros.Keep learning
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