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Multiplication Rule (Independent) Calculator

Rule for finding the probability of two independent events both occurring.

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Probability of A and B

Formula first

Overview

The multiplication rule for independent events provides a method to calculate the joint probability of two events occurring together by multiplying their separate probabilities. It is a fundamental axiom in probability theory that applies only when the occurrence of one event has no impact on the probability of the other.

Symbols

Variables

P(A) = Probability of A, P(B) = Probability of B, P(A B) = Probability of A and B

P(A)
Probability of A
Variable
P(B)
Probability of B
Variable
Probability of A and B
Variable

Apply it well

When To Use

When to use: Use this equation when you need to find the probability of multiple conditions being met simultaneously (the intersection of events). It requires that the events are strictly independent, meaning the outcome of the first event does not change the likelihood of the second.

Why it matters: This rule allows for the modeling of complex systems, such as predicting the reliability of multi-component engineering systems or calculating the odds of genetic inheritance. It is essential in fields like statistics and data science for building predictive models based on discrete variables.

Avoid these traps

Common Mistakes

  • Multiplying probabilities for dependent events without adjustment.
  • 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 fair coin is flipped and a standard six-sided die is rolled. If the probability of getting heads is 0.5 and the probability of rolling a four is 0.16666666666666666, what is the probability of both events occurring?

Probability of A0.5
Probability of B0.16666666666666666

Solve for: pAandB

Hint: Multiply the individual probabilities of the coin flip and the die roll.

Practice Problem 2

In a logistics center, the probability that a specific package is delayed and the sorting machine breaks down simultaneously is 0.02. If the probability of the machine breaking down is 0.05, and these events are independent, find the individual probability of the package being delayed.

Probability of A and B0.02
Probability of B0.05

Solve for: pA

Hint: Rearrange the formula to isolate pA by dividing the joint probability by pB.

Practice Problem 3

A marksman has a 0.8 probability of hitting a target with any single shot. If he takes two shots and the outcomes are independent, what is the probability that he hits the target with both shots?

Probability of A0.8
Probability of B0.8

Solve for: pAandB

Hint: Since the shots are independent, the probability of both occurring is the product of the probability of each shot.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Wikipedia: Independent events
  2. A First Course in Probability by Sheldon Ross, 8th Edition
  3. Probability and Statistics for Engineering and the Sciences by Jay L. Devore, 9th Edition
  4. Wikipedia: Probability
  5. Britannica: Probability
  6. Ross, Sheldon M. A First Course in Probability.
  7. Britannica: Probability theory
  8. AQA GCSE Maths — Probability