In probability, “or” means you want the chance that at least one of two events happens. If you ask “what is the probability of A or B?”, you’re asking for the probability that A happens, B happens, or both happen. This is called the union of two events, written with the symbol ∪, and it comes with a specific formula to calculate correctly.
How “Or” Differs From Everyday Language
In daily conversation, “or” usually means one thing or the other but not both. “Would you like coffee or tea?” implies you’re picking one. In probability and mathematics, “or” is inclusive: it means one, the other, or both. When you calculate P(A or B), you’re capturing every outcome where at least one of the two events occurs. If both happen at the same time, that counts too.
This distinction matters because it changes how you set up calculations. If you only wanted the probability of exactly one event happening (but not both), you’d need a different approach called exclusive or. The standard “or” in every probability textbook and formula is the inclusive version.
The Addition Rule Formula
The core formula for “or” in probability is:
P(A or B) = P(A) + P(B) − P(A and B)
The logic is straightforward. You start by adding the individual probabilities of each event. But if the two events can happen at the same time, some outcomes get counted twice: once when you count A’s outcomes and again when you count B’s. Subtracting P(A and B) corrects for that double counting.
Suppose you’re rolling a single die and you want the probability of rolling an even number or a number greater than 4. The even numbers are 2, 4, and 6, so P(even) = 3/6. The numbers greater than 4 are 5 and 6, so P(greater than 4) = 2/6. But 6 is both even and greater than 4, so P(even and greater than 4) = 1/6. Plugging into the formula: 3/6 + 2/6 − 1/6 = 4/6, or 2/3. The qualifying outcomes are 2, 4, 5, and 6, which is indeed four out of six.
When Events Can’t Happen Together
Sometimes two events are mutually exclusive, meaning they can’t both occur at the same time. Rolling a 2 and rolling a 5 on a single die, for example. When that’s the case, P(A and B) equals zero, and the formula simplifies to:
P(A or B) = P(A) + P(B)
You can only use this shortcut when you’re certain the two events have no overlap. If there’s any chance they can occur together, you need the full formula with the subtraction. A common mistake in introductory courses is applying the simple version when the events do overlap, which inflates the result by counting shared outcomes twice.
When Events Are Independent
Independence is a separate concept from mutual exclusivity, and the two are often confused. Two events are independent when the occurrence of one doesn’t affect the probability of the other. Flipping a coin and rolling a die, for instance. For independent events, you calculate the overlap as:
P(A and B) = P(A) × P(B)
You then plug that product into the general addition rule. Say you flip a coin and roll a die, and you want the probability of getting heads or rolling a 6. P(heads) = 1/2, P(six) = 1/6, and since the coin and die don’t influence each other, P(heads and six) = 1/2 × 1/6 = 1/12. So P(heads or six) = 1/2 + 1/6 − 1/12 = 7/12.
Notice that independent events can absolutely happen at the same time. You can flip heads and roll a 6 on the same trial. That’s why independent events are generally not mutually exclusive, and you still need to subtract the overlap.
Visualizing “Or” With a Venn Diagram
A Venn diagram makes the “or” concept immediately visual. Draw two overlapping circles inside a rectangle. Each circle represents one event, and the rectangle represents all possible outcomes. The “or” of the two events is the entire shaded region covering both circles, including the overlap in the middle. The “and” of the two events is only the overlapping sliver where the circles intersect.
For mutually exclusive events, the two circles don’t overlap at all. The “or” region is simply both circles combined, which is why you can just add the probabilities directly. For events that share outcomes, the overlapping zone shows you exactly what gets double counted and needs to be subtracted.
A Card Deck Example
Say you draw one card from a standard 52-card deck and want the probability of drawing a heart or a king. There are 13 hearts and 4 kings, but the king of hearts is both a heart and a king. Using the addition rule: 13/52 + 4/52 − 1/52 = 16/52, which simplifies to 4/13, roughly 30.8%.
Now change the question to drawing a heart or a diamond. These are mutually exclusive because no card is both a heart and a diamond. So you simply add: 13/52 + 13/52 = 26/52 = 1/2. Exactly half the deck qualifies, which makes intuitive sense since you’re combining two of the four suits.
Extending “Or” Beyond Two Events
The addition rule extends to three or more events, though it gets more complex. For three events A, B, and C:
P(A or B or C) = P(A) + P(B) + P(C) − P(A and B) − P(A and C) − P(B and C) + P(A and B and C)
The pattern follows the same logic. You add individual probabilities, subtract the pairwise overlaps to correct for double counting, then add back the triple overlap because it was subtracted too many times. For mutually exclusive events where no two can happen together, all the overlap terms are zero and you simply add everything up.
A useful shortcut when you need the probability of at least one success across many independent trials is the complement approach: P(at least one) = 1 − P(none). For example, if you roll a die 4 times and want the probability of getting at least one 6, it’s easier to calculate 1 − (5/6)⁴ = 1 − 625/1296 ≈ 51.8% rather than trying to add up every possible combination of one, two, three, or four sixes.