Probability,
with the math shown.

Field guide

Independent vs mutually exclusive, the difference that trips everyone up.

Two of the most-confused terms in introductory probability are independent and mutually exclusive. They sound similar, they both describe how two events relate, and they get swapped in essays all the time. They are not the same and the formula you use depends entirely on which one you have.

Independent events

Two events A and B are independent if knowing whether one happened tells you nothing about the other. Flipping a coin and rolling a die: the coin doesn't care. Drawing a card, putting it back, and drawing again: the deck is reset, so the second draw is independent of the first.

Multiplication is the giveaway. If two events are independent, their joint probability is the product of their individual probabilities. Full stop. The union formula has the multiplicative term subtracted to avoid double-counting the overlap.

Mutually exclusive events

Two events are mutually exclusive (sometimes "disjoint") if they cannot both happen. Rolling a 3 and rolling a 5 on the same die throw: impossible. Drawing a card that is both a heart and a spade: impossible. The intersection is empty by definition.

No multiplication anywhere. The union is just the sum, because there is no overlap to subtract.

Why they are not the same

Here's the test: if two events have any positive probability of both occurring, they are not mutually exclusive. And if two events are mutually exclusive (and both have non-zero probability), they cannot be independent; knowing A happened tells you definitively that B did not, which is the opposite of independence.

• Coin and die. "Heads" and "rolling a 6" are independent (the coin has no idea what the die does) butnot mutually exclusive (you can get both).
• Single die roll. "Rolling a 3" and "rolling a 5" are mutually exclusive (one die roll, one outcome) butnot independent (knowing it's a 3 makes the probability of 5 zero, not 1/6).
• Drawing a card. "Heart" and "Queen" are independent in a single draw; P(Heart) = 1/4, P(Queen) = 1/13, P(Queen of Hearts) = 1/52 = 1/4 × 1/13. They are not mutually exclusive: the Queen of Hearts exists.

Dependent events (when neither label applies)

Most real-world events are neither perfectly independent nor perfectly mutually exclusive; they are simply dependent, in some other way. Drawing two cards without replacement is the canonical example: after the first card leaves the deck, the second draw's probabilities shift. For dependent events you typically need a conditional probability, written P(B | A) ("the probability of B given A"), and the joint formula is:

When P(B | A) = P(B), the events collapse to independent. When P(B | A) = 0, they are mutually exclusive. Independence and mutual exclusivity are the two extremes of the same conditional spectrum.

The three formulas at a glance

• Single event: P(A); complement P(¬A) = 1 − P(A).
• And, independent: P(A ∩ B) = P(A) · P(B).
• Or, independent: P(A ∪ B) = P(A) + P(B) − P(A) · P(B).
• Or, mutually exclusive: P(A ∪ B) = P(A) + P(B).
• And, dependent: P(A ∩ B) = P(A) · P(B | A); calculator above currently handles the independent case for AND; switch to OR if you need the mutually-exclusive variant.

Worked examples

Two coin flips, both heads. Each flip is independent and P(H) = 0.5. So P(HH) = 0.5 · 0.5 = 0.25. Set the calculator to "A and B" with P(A) = P(B) = 0.5 to confirm.

Drawing a heart or a spade in one card draw. Both have probability 1/4 = 0.25, and a single card cannot be both, so they are mutually exclusive. P(heart ∪ spade) = 0.25 + 0.25 = 0.5. Set the calculator to "A or B" with the relation toggle on mutually exclusive.

Rain on Saturday or Sunday (treated as independent). P(rain Sat) = 0.30, P(rain Sun) = 0.40. Probability it rains at all over the weekend: 0.30 + 0.40 − (0.30 · 0.40) = 0.58. The calculator's default state shows exactly this.

Reading the output (decimal, percent, fraction)

A probability of 0.58 means roughly 58 favorable outcomes out of every 100 and the fraction view (29/50) gives the cleanest exact form. The fraction is computed via continued-fraction expansion, which returns the best rational approximation with a denominator ≤ 10,000. For inputs like 0.5, 0.25, or 0.333… you'll get the textbook fractions 1/2, 1/4, and 1/3.

Common mistakes

• Adding probabilities without subtracting the overlap. If A and B can co-occur, P(A) + P(B) double-counts the cases where both happen.
• Treating "or" as automatically additive. It is, but only if the events are mutually exclusive. Otherwise subtract the intersection.
• Confusing "independent" with "unrelated." Two events can be causally unrelated and still not independent in the probability sense (consider sampling without replacement).
• Ending with a probability greater than 1 or less than 0. If you get one, an input is wrong; probabilities are always in [0, 1].

Disclaimer

This calculator handles independent and mutually-exclusive cases for two events. For more complex scenarios, such as dependent sequences, Bayes' rule, and conditional probabilities, you'll want a stats package or a worked-out conditional table.