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Probability of an Impossible Event
The probability of an impossible event is always 0 as for an impossible event, E = 0 and thus, P(E) = 0. Probability is defined as the possibility of an event to occur which can be calculated as the ratio of the number of favorable outcomes to the total number of outcomes of an event.
In this article, let's discuss the probability of an impossible event with solved examples and practice questions.
1.  What is the Probability of an Impossible Event? 
2.  Finding the Probability of an Impossible Event 
3.  Probability of an Impossible Event Examples 
4.  FAQs on Probability of an Impossible Event 
What is the Probability of an Impossible Event?
Probability measures the likelihood of occurrence of an event in numeric terms. We come across many reallife situations where we predict the outcome of an event. We may either be sure or not sure of the results of an event. A sure event, as the name suggests, is the one that is confirmed to happen. For example, while tossing a coin, the outcome will definitely be either head or tail, known as the sure event. Thus, the probability of a sure event is 1 and the probability of an impossible event is 0.
The probability of an impossible event refers to the probability of an event not happening at all and there's no possibility of it be happening in the future. Thus, the probability of such an event is always equal to 0.
Finding the Probability of an Impossible Event
In order to find the probability of any event, we consider the total number of outcomes as well the number of favorable outcomes. Let us take an example to understand it better. If we roll a fair die, the possibility is that you will get a number not more than 6. There is a likelihood of getting 1, 2, 3, 4, 5, or 6. Here, outcome 9 or any number greater than 6 is an impossible outcome, not included in sample space. Thus, we can conclude that the probability of an impossible event is 0.
Does the Probability of an Impossible Event mean Empty Set?
One of the basic theorems of probability states that the probability of an impossible event is phi or an empty set. Also, P(ϕ) = 0.
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Important Notes
Let us summarize the key learnings for the Probability of an Impossible Event.
 The probability of an impossible event is the likelihood of an impossible event, that is the event that has zero possibility of happening.
 Probability is represented as a fraction or a ratio of favorable outcomes to the total number of outcomes and always lies between 0 and 1.
 An impossible event is not a subset of sample space.
Probability of an Impossible Event Examples

Example 1: Out of 26 English alphabet cards, determine the probability of getting a card with the number 8.
Solution:
Given: 26 cards (English alphabets)
To find: The probability of getting a card with the number 8.
Total Outcomes = 26
Getting a card with the number 8 is an impossible event.
Thus, favourable outcomes = 0 or P(E) = 0
Therefore, the probability of the impossible outcome, that is, getting a card with the number 8 out of 26 alphabet cards = 0.

Example 2: What is the probability of getting 5 aces from a deck of 52 cards.
Solution:
Given: 52 cards (deck of cards)
To find: The probability of getting 5 aces.
Total Outcomes = 52
Getting 5 aces from a deck of 52 cards is an impossible event.
Thus, favourable outcomes = 0 or P(E) = 0
Therefore, the probability of the impossible outcome, that is, getting 5 aces from a deck of 52 cards = 0.
FAQs on Probability of an Impossible Event
What Is the Probability of an Impossible Event in Statistics?
In statistics, the probability of an impossible event is equal to 0. For an impossible event, E = 0 and thus, P(E) = 0. For example, the probability of drawing a green ball, out of a set of red balls is zero as getting a green ball when you just have red balls in the set, is an impossible event.
How To Find the Probability of an Impossible Event?
Probability is a numeric measure of the possibility of the occurrence of an event. For any event, we are either sure or not sure. Since the sure event confirms the possibility of an event to occur, its probability is 1, whereas, an event that is not likely to happen at all is an impossible event, its probability is 0.
Can the Probability of an Impossible Event be a Null Set?
An event that will never occur is referred to as an impossible event. Thus, we can conclude that an impossible event has a probability of zero, that is P(E) = 0. Also, P(ϕ)=0. Therefore, the probability of an impossible event is a null set.
Does the Probability of an Impossible Event Mean Zero Set?
The events that are impossible to happen have zero probability, but its converse is not necessarily true. This is clear that an empty set has zero probability but a zero probability event does not signify that an event is an impossible event.
What Is the Probability of an Impossible Event and a Sure Event?
A sure event is the one that is definitely going to happen for sure, whereas, an impossible event has no possibility of its occurrence. Thus, the probability of a sure event is 1 and the probability of an impossible event is 0.
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