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Mutually Exclusive Events

TL;DR

Section titled “TL;DR”
  • Two events are mutually exclusive when the occurrence of one prevents the occurrence of the other.
  • Simple examples: a coin landing heads vs tails; a die landing a 3 vs landing a 4, 5, or 6.
  • Used when determining event likelihoods (e.g., P(heads) = 1/2, P(3) = 1/6) and when choosing between exclusive options.

Definition

Section titled “Definition”

Mutually exclusive events, also known as disjoint events, are events that cannot happen at the same time. In other words, if one event occurs, the other cannot. These events are typically represented by the symbol ”∩” in probability theory, indicating that they intersect or overlap.

Explanation

Section titled “Explanation”

If two events are mutually exclusive, they have no simultaneous occurrence: the occurrence of one rules out the occurrence of the other. The concept is illustrated by simple chance experiments where only one outcome can happen at a time. Mutually exclusive events are relevant for calculating probabilities of specific outcomes and for decision-making situations where options cannot both be selected.

Examples

Section titled “Examples”

Coin flip

Section titled “Coin flip”

The two possible outcomes of flipping a coin are heads or tails. If the coin lands on heads, it cannot also land on tails, and vice versa. Therefore, the events of flipping a heads and flipping a tails are mutually exclusive. For a fair coin, the probability of flipping heads is 1/2.

Die roll

Section titled “Die roll”

A die has six sides, each with a different number. If the die is rolled and lands on a 3, it cannot also land on a 4, 5, or 6. Therefore, the events of rolling a 3 and rolling a 4, 5, or 6 are mutually exclusive. For a fair six-sided die, the probability of rolling a 3 is 1/6.

Decision example

Section titled “Decision example”

If a person is deciding whether to go to the beach or the mountains for their vacation, they cannot do both at the same time. Therefore, the events of going to the beach and going to the mountains are mutually exclusive; the person must choose one event over the other.

Use cases

Section titled “Use cases”
  • Calculating probabilities for outcomes that cannot occur together (examples above).
  • Modeling exclusive choices in decision-making scenarios (e.g., mutually exclusive vacation options).
Section titled “Related terms”
  • Disjoint events
  • Intersection (∩)
  • Probability