Impossible Event
Key Formula
P(E)=0
Where:
- P(E) = The probability of the impossible event E
- 0 = Zero, indicating the event can never happen
Worked Example
Problem: A bag contains 4 red marbles and 6 blue marbles. What is the probability of drawing a green marble from the bag?
Step 1: Identify the sample space. The bag contains only red and blue marbles, so the possible outcomes when drawing one marble are: red or blue.
S={red, blue}
Step 2: Count the number of favorable outcomes. There are zero green marbles in the bag, so the number of ways to draw a green marble is 0.
Favorable outcomes=0
Step 3: Calculate the probability using the formula: favorable outcomes divided by total outcomes.
P(green)=100=0
Answer: The probability of drawing a green marble is 0. This is an impossible event because no green marbles exist in the bag.
Another Example
Problem: You flip a fair coin once. What is the probability that the coin lands on both heads and tails at the same time?
Step 1: Identify the sample space. A single coin flip has two possible outcomes.
S={heads, tails}
Step 2: Determine whether the event 'both heads and tails simultaneously' exists in the sample space. A coin can land on only one side per flip, so this outcome is not in the sample space.
E=∅
Step 3: The event corresponds to the empty set, which contains no outcomes.
P(E)=P(∅)=0
Answer: The probability is 0. Landing on both heads and tails simultaneously is an impossible event.
Frequently Asked Questions
Is an event with probability 0 always impossible?
In finite sample spaces (like rolling dice or drawing cards), yes — probability 0 means the event cannot happen. However, in advanced probability with infinite sample spaces, an event can have probability 0 yet still be theoretically possible. For standard school-level problems with a finite number of outcomes, you can treat probability 0 and impossible as the same thing.
What is the difference between an impossible event and an unlikely event?
An impossible event has a probability of exactly 0 and can never occur. An unlikely event has a probability greater than 0 but close to it, meaning it could happen but rarely does. For instance, rolling a 7 on a standard die is impossible (probability 0), while rolling three sixes in a row is unlikely (probability 1/216 ≈ 0.005) but entirely possible.
Impossible Event vs. Sure Event
An impossible event has a probability of 0 and can never occur; a sure event (also called a certain event) has a probability of 1 and always occurs. They are exact opposites on the probability scale. If event E is impossible, then its complement (everything other than E) is a sure event. For example, rolling a 7 on a standard die is impossible, while rolling a number between 1 and 6 is certain.
Why It Matters
Understanding impossible events helps you set the lower boundary of probability. Every probability value falls between 0 and 1, and the impossible event anchors that scale at zero. Recognizing impossible events also prevents wasted effort in real problems — if you can identify that an outcome is impossible, you can eliminate it immediately and focus on the events that actually contribute to your calculations.
Common Mistakes
Mistake: Confusing very unlikely events with impossible events.
Correction: An event is only impossible if its probability is exactly 0. Winning a lottery is extremely unlikely but not impossible — someone eventually wins. Reserve the term 'impossible' for events that truly cannot happen, like drawing a king of diamonds from a deck that contains no king of diamonds.
Mistake: Thinking the empty set and the impossible event are different things.
Correction: In probability, the impossible event is represented by the empty set (∅). They are the same concept. The empty set contains no outcomes, which is precisely why its probability is 0.
Related Terms
- Probability — The measure that equals 0 for impossible events
- Sure Event — The opposite: an event with probability 1
- Zero — The probability value of any impossible event
- Sample Space — The set of all possible outcomes
- Empty Set — The set that represents an impossible event
- Complement — Complement of impossible event is certain