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Complementary Probability Calculator

Rule for finding the probability of an event NOT occurring.

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Probability of not A

Formula first

Overview

The complementary probability rule defines the relationship between the likelihood of an event occurring and the likelihood of it not occurring. It is derived from the fundamental probability axiom that the sum of all mutually exclusive and exhaustive outcomes in a sample space must equal one.

Symbols

Variables

P(A) = Probability of A, P( A) = Probability of not A

P(A)
Probability of A
Variable
Probability of not A
Variable

Apply it well

When To Use

When to use: Use this formula when calculating the probability of the 'opposite' outcome is simpler than calculating the outcome itself. It is the standard approach for solving 'at least one' problems or binary scenarios where only two states are possible.

Why it matters: This principle allows for significant computational efficiency in risk management and insurance, where finding the probability of success is often easier than tallying every possible mode of failure. It ensures that statistical models remain consistent and bound within the 0 to 1 range.

Avoid these traps

Common Mistakes

  • Forgetting to subtract from 1.
  • Convert units and scales before substituting, especially percentages, time units, or powers of ten.
  • Interpret the answer with its unit and context; a percentage, rate, ratio, and physical quantity do not mean the same thing.

One free problem

Practice Problem

Practice Problem 1

A local meteorologist predicts that there is a 35% chance of rain today. What is the probability that it will not rain today?

Probability of A0.35

Solve for: pNotA

Hint: Subtract the probability of the event (rain) from the total certainty of 1.

Practice Problem 2

In a high-precision manufacturing plant, the probability of a specific component being defective is 0.004. Calculate the probability that a randomly selected component is functional (not defective).

Probability of A0.004

Solve for: pNotA

Hint: The sum of the defective rate and the functional rate must equal 1.

Practice Problem 3

An analyst determines that the probability of a stock price not increasing tomorrow is 0.82. Based on this, what is the probability that the stock price will increase?

Probability of not A0.82

Solve for: pA

Hint: Rearrange the formula to solve for P(A) by subtracting the complement from 1.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Probability
  2. Wikipedia: Complementary event
  3. A First Course in Probability by Sheldon Ross
  4. Sheldon Ross, A First Course in Probability
  5. Wikipedia: Probability axioms
  6. Ross, Sheldon M. A First Course in Probability. 8th ed. Pearson, 2010.
  7. Edexcel GCSE Maths — Probability