## What is the Probability of an Impossible Event?

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

*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]:

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].

## References

[1] Probability: Key Definitions. Retrieved June 11, 2020 from: https://www.regent.edu/app/uploads/2018/06/ML-Math-101-Probability.pdf” rel=”noopener” target=”_blank”>Probability: Key Definitions. Retrieved June 11, 2020 from: https://www.regent.edu/app/uploads/2018/06/ML-Math-101-Probability.pdf

[2] Probability I. Retrieved June 11, 2021 from: https://www.radford.edu/~scorwin/courses/200/book/90ProbabilityI.html