Probability Calculator

Calculate basic event probability, combined probabilities (AND/OR), and binomial probability. All results show step-by-step solutions.

P(event) = favorable / total
P = 1 / 6 = 0.1667 = 16.67%

Events are independent:
  P(A and B) = P(A) × P(B)
  P(A and B) = 0.5 × 0.3 = 0.15

  P(A or B) = P(A) + P(B) - P(A and B)
  P(A or B) = 0.5 + 0.3 - 0.15 = 0.65

  P(not A) = 1 - 0.5 = 0.5
  P(not B) = 1 - 0.3 = 0.7

Binomial probability formula:
  P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Values: n = 10, k = 3, p = 0.5, q = 1-p = 0.5

Step 1 — Calculate C(10, 3):
  C(10, 3) = 10! / (3! × 7!)
  = (10 × 9 × 8) / (1 × 2 × 3)
  = 120

Step 2 — Calculate p^k = 0.5^3:
  = 0.125

Step 3 — Calculate (1-p)^(n-k) = 0.5^7:
  = 0.007813

Step 4 — Multiply:
  P(X = 3) = 120 × 0.125 × 0.007813
  P(X = 3) = 0.117188

What Is Probability?

Probability is the mathematical study of chance. It quantifies how likely an event is to occur on a scale from 0 (impossible) to 1 (certain). For a simple experiment where all outcomes are equally likely:

P(event) = favorable outcomes / total outcomes

For example, when rolling a fair six-sided die, the probability of rolling a 4 is 1/6 ≈ 0.1667 (16.67%), because there is 1 favorable outcome (rolling a 4) out of 6 total equally likely outcomes.

Basic Probability Rules

The Complement Rule

The probability that an event does NOT occur is 1 minus the probability that it does:

P(not A) = 1 − P(A)

Example: The probability of drawing an ace from a standard deck is 4/52 = 1/13. The probability of NOT drawing an ace is 1 − 1/13 = 12/13 ≈ 0.9231.

The Addition Rule: P(A or B)

The probability that event A or event B (or both) occur is:

P(A or B) = P(A) + P(B) − P(A and B)

Subtracting P(A and B) prevents double-counting outcomes where both events happen. For mutually exclusive events (events that cannot both occur), P(A and B) = 0, so the formula simplifies to P(A or B) = P(A) + P(B).

Example: Drawing a heart or a face card from a standard deck. P(heart) = 13/52. P(face card) = 12/52. P(heart and face card) = 3/52 (J♥, Q♥, K♥). P(heart or face card) = 13/52 + 12/52 − 3/52 = 22/52 ≈ 0.4231.

The Multiplication Rule: P(A and B)

The probability that both event A and event B occur:

P(A and B) = P(A) × P(B)  [if A and B are independent]

For independent events, the outcome of one does not affect the other. For dependent events, use: P(A and B) = P(A) × P(B|A).

Example (independent): Rolling two dice and getting a 6 on both. P = 1/6 × 1/6 = 1/36 ≈ 0.0278.

Example (dependent): Drawing two aces without replacement. P(first ace) = 4/52. P(second ace | first ace drawn) = 3/51. P(both aces) = 4/52 × 3/51 = 12/2652 ≈ 0.0045.

Binomial Probability

Binomial probability applies to a series of repeated independent trials, each with the same probability of success. The binomial formula gives the probability of exactly k successes in n trials:

P(X = k) = C(n, k) × p^k × (1 − p)^{n − k}

Where C(n, k) = n! / (k! × (n − k)!) is the number of ways to choose k successes from n trials (the binomial coefficient, read "n choose k").

Example: A basketball player makes 70% of free throws. What is the probability of making exactly 4 of 5 shots?

n = 5, k = 4, p = 0.7
C(5, 4) = 5! / (4! × 1!) = 5
P = 5 × 0.7⁴ × 0.3¹ = 5 × 0.2401 × 0.3 = 0.3601 (36.01%)

Common Probability Examples

Frequently Asked Questions

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