Probability formulas

Basic probability, complements, the AND and OR rules, permutations, combinations and the binomial formula.

These pages explain formulas for reference. They don't mean Mathover can solve every problem that uses them — interactive calculations are limited to the linked tools.

Basic probability

P(A) = favourable outcomes ÷ total outcomes

Count the outcomes where A happens and divide by all possible outcomes.

Symbols

P(A): probability of event A, between 0 and 1

When it applies

Only when every outcome is equally likely (a fair die, a well-shuffled deck).

Worked example

A fair six-sided die is rolled. Find P(even number).

1.Favourable: 2, 4, 6 — 3 outcomes
2.Total: 6 outcomes
3.P = 3/6 = 1/2

Answer: 1/2

Try it in the Probability Calculator tool →

Complement rule

P(not A) = 1 − P(A)

Either A happens or it doesn't, and those probabilities add to 1.

Symbols

P(A): probability of A
P(not A): probability A does not happen

When it applies

Always, for any event A.

Worked example

The probability of rain tomorrow is 0.3. Find the probability of no rain.

1.P(no rain) = 1 − 0.3
2.= 0.7

Answer: 0.7

Reference only — no interactive Mathover tool for this formula yet.

Independent events (AND)

P(A and B) = P(A) × P(B)

When one event doesn't affect the other, multiply their probabilities.

Symbols

A, B: independent events

When it applies

Only for independent events. For dependent events use P(A) × P(B given A).

Worked example

A fair coin is flipped and a fair die is rolled. Find P(heads and a 6).

1.P(heads) = 1/2, P(6) = 1/6
2.1/2 × 1/6 = 1/12

Answer: 1/12

Try it in the Probability Calculator tool →

Mutually exclusive events (OR)

P(A or B) = P(A) + P(B)

If A and B can't happen at the same time, add their probabilities.

Symbols

A, B: mutually exclusive events

When it applies

Only when A and B cannot both happen. Otherwise use the general addition rule.

Worked example

One card is drawn from a standard 52-card deck. Find P(king or queen).

1.P(king) = 4/52, P(queen) = 4/52
2.4/52 + 4/52 = 8/52 = 2/13

Answer: 2/13

Try it in the Probability Calculator tool →

General addition rule

P(A or B) = P(A) + P(B) − P(A and B)

Adding P(A) and P(B) counts the overlap twice, so subtract it once.

Symbols

P(A and B): probability both happen

When it applies

Any two events. Reduces to the mutually exclusive rule when P(A and B) = 0.

Worked example

One card is drawn from a standard deck. Find P(heart or king).

1.P(heart) = 13/52, P(king) = 4/52
2.P(king of hearts) = 1/52
3.13/52 + 4/52 − 1/52 = 16/52 = 4/13

Answer: 4/13

Reference only — no interactive Mathover tool for this formula yet.

Permutations

ⁿPᵣ = n! ÷ (n − r)!

The number of ordered arrangements of r items chosen from n different items.

Symbols

n: number of items available
r: number chosen
n!: n × (n − 1) × … × 1

When it applies

Order matters, no repetition, 0 ≤ r ≤ n.

Worked example

How many ways can 1st, 2nd and 3rd place be awarded among 5 runners?

1.⁵P₃ = 5! ÷ 2!
2.= 5 × 4 × 3
3.= 60

Answer: 60

Try it in the Probability Calculator tool →

Combinations

ⁿCᵣ = n! ÷ (r!(n − r)!)

The number of ways to choose r items from n when order doesn't matter.

Symbols

n: items available
r: items chosen

When it applies

Order doesn't matter, no repetition, 0 ≤ r ≤ n.

Worked example

How many ways can a committee of 3 be chosen from 10 people?

1.¹⁰C₃ = 10! ÷ (3! × 7!)
2.= (10 × 9 × 8) ÷ (3 × 2 × 1)
3.= 720 ÷ 6 = 120

Answer: 120

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Binomial probability

P(X = k) = ⁿCₖ × pᵏ × (1 − p)ⁿ⁻ᵏ

The probability of exactly k successes in n trials.

Symbols

n: number of trials
k: number of successes
p: probability of success on each trial

When it applies

Fixed number of trials, each trial independent, two outcomes, and the same p every time.

Worked example

A fair coin is flipped 5 times. Find P(exactly 2 heads).

1.⁵C₂ = 10
2.p² = (1/2)² = 1/4, (1 − p)³ = (1/2)³ = 1/8
3.10 × 1/4 × 1/8 = 10/32 = 5/16

Answer: 5/16 = 0.3125

Try it in the Probability Calculator tool →