Probability of an Impossible Event

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What is the Probability of an Impossible Event?

The probability of an impossible event is 0. Impossible events can’t occur.

probability of an impossible event
Rolling a 7 on a six-sided die is an impossible event.
For example: What is the probability of rolling a 7 on a six-sided die? As the number 7 never appears on a face of a six-sided die, the event is impossible. Therefore, the probability is 0.

A certain event is just the opposite: it will happen under every circumstance. For example, if a six-sided die has the number 3 on all faces, then the probability of the event “rolling a 3” is 1.

Zero Probability Doesn’t Always Mean “Impossible”

Although the probability of an impossible event is zero, it doesn’t mean that every event with zero probability is impossible. There are many situations where an event doesn’t happen under some circumstances (i.e. it has a probability of zero for a certain model or situation) but it can happen in others.

Here’s a straightforward example: Given that a classroom contains all boys, what is the probability of choosing a girl at random? The probability is 0, as it’s impossible to choose a girl if the classroom only has boys. However, that doesn’t mean the event of choosing a girl is impossible: change the situation slightly (by adding girls to the classroom), and the probability changes.

Probability of an Impossible Event = Empty Set

An impossible event is equal to the empty set ∅. This is a rule of probability, which can formally be written as [1]:

p (∅) = 0.

The impossible event, defined in this way as a set with no elements, doesn’t correspond to any experimental result, but it is useful in some calculations [2].


[1] Probability: Key Definitions. Retrieved June 11, 2020 from:” rel=”noopener” target=”_blank”>Probability: Key Definitions. Retrieved June 11, 2020 from:
[2] Probability I. Retrieved June 11, 2021 from:

Stephanie Glen. "Probability of an Impossible Event" From Elementary Statistics for the rest of us!

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