In probability, an event is any outcome or collection of outcomes from an experiment. More precisely, it’s a subset of the sample space, which is the full set of everything that could possibly happen. When you roll a die, the sample space is {1, 2, 3, 4, 5, 6}. “Rolling a 4” is an event. “Rolling an even number” is also an event. Both are subsets of that sample space, just different-sized ones.
An event is said to “occur” when the actual outcome of the experiment turns out to be one of the outcomes included in that event. If you defined your event as “rolling an even number” and the die lands on 2, the event occurred. If it lands on 3, it didn’t.
Simple Events vs. Compound Events
A simple event (also called an elementary event) consists of exactly one outcome. Rolling a 4 on a die is a simple event. Drawing the ace of spades from a deck is a simple event. There’s only one way for it to happen.
A compound event includes two or more outcomes from the same sample space. “Rolling a number greater than 4” is a compound event because it includes both 5 and 6. “Drawing a heart” from a standard deck is compound because 13 different cards satisfy it. Compound events are where probability gets more interesting, because you’re grouping outcomes together based on some shared property and asking how likely any of them are to occur.
How Event Probability Is Calculated
Every event has a probability between 0 and 1. A probability of 0 means the event is impossible. Rolling an 8 on a standard six-sided die, for example, gives you 0 out of 6 possible outcomes, so the probability is 0. A probability of 1 means the event is certain. Rolling a value between 1 and 6 on that same die gives you 6 out of 6, so the probability is 1.
For equally likely outcomes, the calculation is straightforward: count the outcomes in your event and divide by the total number of outcomes in the sample space. The probability of rolling an even number on a fair die is 3/6, or 0.5, because three of the six faces (2, 4, 6) are even.
The Complement of an Event
Every event has a complement: everything in the sample space that isn’t part of the event. If your event is “rolling a 6,” the complement is “rolling anything other than a 6.” The probability of an event and its complement always add up to 1. This gives you a useful shortcut: if you know the probability of something happening, you can find the probability of it not happening by subtracting from 1. If there’s a 30% chance of rain, there’s a 70% chance of no rain.
Mutually Exclusive Events
Two events are mutually exclusive when they can’t both happen at the same time. Rolling a 3 and rolling a 5 on a single die roll are mutually exclusive. You’ll get one or the other, never both. Because they can’t overlap, the probability of both occurring together is zero.
This matters when you want the probability of either event happening. With mutually exclusive events, you simply add their individual probabilities. The chance of rolling a 3 or a 5 is 1/6 + 1/6 = 2/6. But if events can overlap (like “rolling an even number” and “rolling a number less than 4,” which share the outcome 2), you need to subtract the overlap to avoid counting it twice.
Independent vs. Dependent Events
Two events are independent if the outcome of one has no effect on the probability of the other. Flipping a coin and then rolling a die are independent. The coin doesn’t care what the die does. For independent events, the probability of both occurring is found by multiplying their individual probabilities.
Dependent events are the opposite: the outcome of the first changes the probability of the second. Drawing two cards from a deck without replacing the first is a classic example. If you draw an ace first, there are now only 3 aces left among 51 remaining cards instead of 4 among 52. The first draw changed the landscape for the second.
Exhaustive Events
A set of events is collectively exhaustive when, taken together, they cover every possible outcome in the sample space. At least one of them must occur. On a die roll, the events “rolling 1, 2, or 3” and “rolling 4, 5, or 6” are collectively exhaustive because there’s no outcome they miss. If your events are both mutually exclusive and collectively exhaustive, you’ve split the entire sample space into non-overlapping pieces with no gaps, which is a common setup in probability problems.
Standard Notation
Probability uses a compact set of symbols borrowed from set theory. Events are labeled with capital letters like A, B, and C. “A or B” (at least one of them happens) is written as A ∪ B, the union. “A and B” (both happen) is written as A ∩ B, the intersection. “Not A” (the complement) is typically written as Aᶜ. The probability of event A is written P(A).
These symbols let you express complex scenarios concisely. P(A ∪ B) asks for the probability that at least one of two events occurs. P(Aᶜ) asks for the probability that A does not occur, which equals 1 minus P(A).
Events in Real-World Probability
Outside of textbook dice and cards, events are defined the same way, just with messier sample spaces. In insurance, analysts define events like “a policyholder files a claim this year” or “a borrower’s payment is 30 or more days late.” Each of these is a subset of all possible outcomes for that person’s financial behavior in a given time period. Medical researchers define events such as “a patient screens positive for depression” or “a diagnostic test returns a false positive.” The framework is identical: you specify the outcome or set of outcomes you care about, then assign or estimate a probability.
What makes probability powerful is that once you define your events precisely, the same rules for complements, independence, and mutual exclusivity apply whether you’re rolling dice or modeling whether someone will develop a disease. The math doesn’t change. Only the complexity of figuring out the actual probabilities does.