1145 LearnersLast updated on June 18th, 2025
Probability of an Impossible Event
Probability measures how likely an event is to occur. Some events are very likely, while others can be highly unlikely. An impossible event is an event that has no chance of happening at all. For example, rolling a 0 on a six-sided die. In this topic, we are going to talk about the probability of an impossible event.
What is the Probability of an Impossible Event?
An impossible event has no chance of happening at all no matter how many times we perform the event. This is why in probability theory, we assign a probability of 0 to impossible events because the event will never occur.
P (Impossible Event) = 0
A simple example would be drawing a black card from a deck of all red cards. Imagine there is a deck of 52 cards, and you remove all the black cards, leaving only the red cards. Now, if a card is randomly picked from this deck, what is the probability of getting a black card? The answer is 0. There is no chance of getting a black card because all black cards have been removed. So the probability of getting a black card in the deck of red cards is 0.
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How to Find the Probability of an Impossible Event?
To find the probability of any event, divide the number of favorable outcomes by the total number of possible outcomes. When there is an event with an outcome that is impossible to include in the sample space, we can say that this is an impossible event and the probability of it happening is 0
In probability theory, an impossible event corresponds to a null set, and its probability is always 0.
Real-life Applications of the Probability of an Impossible Event
Determining the probability of impossible events is used in research or manufacturing. Here are a few real-world applications:
Cybersecurity:
The probability of an impossible event applies when blocking unauthorized access due to incorrect credentials.
Traffic regulation:
Some vehicles only start if the seatbelt is fastened. The probability of driving in these cars without the seatbelt fastened is 0.
Lotteries:
Most lottery systems are programmed to accept valid number ranges. If a person selects a number that is outside the number range, the probability of winning is 0.
Common Mistakes and How to Avoid Them in Probability of an Impossible Event
When dealing with the probability of impossible events students may make mistakes. So here are some common mistakes that students make and ways to avoid them:
Mistake 1
Assuming that an impossible event will never be mentioned in a problem
Students assume that because an event is impossible it will not be included in a probability question. Make sure to check whether the event is logical before assigning its probability.
Mistake 2
Confusing impossible events with unlikely events
When finding the probability of events, students may assume that an unlikely event is the same as an impossible event. This is not correct, an unlikely event has a small probability of happening, whereas an impossible event has no chance of happening at all.
Mistake 3
Assuming that probability can be negative
Thinking that an impossible event can have a probability of -1 or -0.5 is wrong. Probabilities are always between 0 and 1, inclusive, and cannot be negative.
Mistake 4
Getting impossible and mutually exclusive events mixed up
Students may assume that mutually exclusive events (events that cannot happen together) are the same as impossible events. Mutually exclusive events cannot happen at the same time and occur separately, but impossible events never occur at all.
Mistake 5
Thinking that all events with a probability of 0 are impossible events.
Some events have a probability of 0, but that does not mean they are impossible. This mainly occurs in continuous probability.
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Solved examples on the Probability of an Impossible Event
Problem 1
What is the probability of rolling a 7 on a standard six-sided die?
0
Explanation
A standard die has only 6 numbers (1 - 6).
Since 7 is a number not present on the die, this event is impossible,
so P (rolling 7) = 0
Problem 2
What is the probability of selecting a blue card from a standard 52-card deck?
0
Explanation
A standard deck has 52 cards
Total outcomes = 52 cards (all cards in the deck)
Favorable outcomes = 0 (as no card is blue)
Since this is an impossible event, P (getting a blue card) = 0
Problem 3
What is the probability of picking a negative number from the set {1, 2, 3, 4, 5}?
0
Explanation
The given set is {1, 2, 3, 4, 5}
Total outcomes: 5
Favorable outcomes = 0 (since there are no negative numbers)
Since this is an impossible event, P(picking a negative number) = 0
Problem 4
What is the probability that the square root of 10 is an integer?
0
Explanation
The square root of 10 is an irrational number, so it can never be an integer.
Problem 5
A student takes a 10-question true/false test. What is the probability of scoring 11 correct answers?
0
Explanation
Total outcomes = 10
Favorable outcomes = 0 (because the max score is 10)
This is an impossible event because P (scoring 11 correct answers ) = 0
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FAQs on the Probability of an Impossible Event
1.What is meant by impossible events?
2.What is the probability of an impossible event?
3.If an event has a probability of 0, can it still happen?
4.Why is a probability of 0 assigned to impossible events?
5.What is a null set in impossible events?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
