Mutually Exclusive Events

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Mutually Exclusive Events: meaning, method and solved example (Algebra 2) – Maths PDF

In short: Mutually exclusive events are outcomes that cannot occur at the same time in one trial, so their overlap is empty and the chance of either one is just the sum of their chances.

Algebra 2ProbabilityS-CP.B.7

What is Mutually Exclusive Events?

Mutually exclusive events are like the two sides of a coin or the colors of a single marble: if one happens, the other is ruled out. Their regions on a diagram never touch.

Events \(A\) and \(B\) are mutually exclusive when \(P(A \text{ and } B) = 0\). Then \(P(A \text{ or } B) = P(A) + P(B)\). They are also called disjoint events. Note that exclusive events with positive probability are never independent, because knowing one occurred tells you the other did not.

How to spot it

  • A single draw, roll or spin is made, and the outcomes named in the problem cannot both be true, which marks mutually exclusive events.
  • The word or links two events that describe different colors, numbers or categories of the same pick.
  • A Venn diagram would show two circles that do not overlap.
  • A question asks whether it is possible for both events to happen in one trial.

The method

  1. List the outcomes that belong to each event for a single trial.
  2. Check for an outcome in both lists; if there is none, you have mutually exclusive events.
  3. Find each probability as favorable outcomes over total outcomes.
  4. Add the two probabilities to get \(P(A \text{ or } B)\), and subtract from 1 for the chance of neither.
red12 of 40blue9 of 40all 40 tilesno overlap: P(red or blue) = 21/40
Red and blue never overlap, so the probabilities add: 12/40 + 9/40 = 21/40.

Mutually Exclusive Events: solved example

Problem

A bag holds 40 tiles: 12 red, 9 blue, 7 green and 12 yellow. One tile is drawn. Find the probability that it is red or blue, and the probability that it is not green.

Solution

One tile cannot be both red and blue, so the events are mutually exclusive: \(P(\text{red or blue}) = \frac{12}{40} + \frac{9}{40} = \frac{21}{40}\).

For not green, use the complement: \(1 – \frac{7}{40} = \frac{33}{40}\). The answers are \(\frac{21}{40}\) and \(\frac{33}{40}\).

Try it

Question 1

A die is rolled once. Are “even” and “5” mutually exclusive? Find \(P(\text{even or }5)\).

Show the answer

Yes, no number is both. The favorable outcomes are 2, 4, 5, 6, so the probability is \(\frac{4}{6} = \frac{2}{3}\).

Question 2

Events \(A\) and \(B\) are mutually exclusive with \(P(A) = 0.35\) and \(P(B) = 0.4\). Find \(P(A \text{ or } B)\) and \(P(\text{neither})\).

Show the answer

\(0.35 + 0.4 = 0.75\), and neither has probability \(1 – 0.75 = 0.25\).

Mutually Exclusive Events vs. Independent Events

Exclusive events cannot both happen, so one occurring makes the other impossible. Independent events can both happen, and one occurring does not change the other’s chance. Drawing a red tile and then a second tile with replacement is independent; drawing red and blue in a single draw is exclusive.

Independent Events

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