Introduction to Probability

What is the addition rule in probability?

Quick answer

The addition rule states that the probability of event A or event B occurring equals the sum of their individual probabilities minus the probability that both occur together: P(A or B) = P(A) + P(B) - P(A and B). When two events are disjoint, they share no outcomes, so P(A and B) is zero and the rule simplifies to P(A) + P(B).

The topic explained

Union, intersection, and the addition rule

In probability, the union of two events means that at least one of them occurs. We write this with the word 'or'. The intersection means both events occur at the same time, written with the word 'and'.

The addition rule in probability links these ideas together. It states that P(A or B) = P(A) + P(B) - P(A and B). You add the individual probabilities of each event, then subtract the probability of the overlap.

Why you subtract the intersection

When you add P(A) and P(B), any outcome that belongs to both events gets counted twice. Subtracting the intersection restores the balance so that shared outcomes are counted exactly once.

Imagine a class of 20 students where 10 play soccer, 9 play basketball, and 4 play both sports. If you add 10 and 9, you get 19, which counts the 4 dual-sport athletes twice. The actual number of students playing at least one sport is 10 + 9 - 4 = 15. The probability of choosing one of these students is 15/20, or 3/4.

When two events are disjoint

Two events are disjoint (also called mutually exclusive) when they share no outcomes. They cannot happen at the same time. For instance, rolling a standard die cannot yield an odd number and an even number simultaneously.

When events are disjoint, their intersection is empty, meaning P(A and B) = 0. In this special case, the formula simplifies to P(A or B) = P(A) + P(B). To test if events are disjoint, look for a single shared outcome. If you find even one, they are not disjoint and you must subtract the intersection.

Common pitfalls to avoid

Watch out for the word 'or'. In everyday speech, 'or' often suggests one choice or the other, but not both. In mathematics, 'or' is inclusive and includes the outcome where both happen.

Do not assume events are disjoint just because their descriptions sound different. On a six-sided die, rolling an even number and rolling a number greater than 4 seem like distinct conditions, but 6 satisfies both. Always inspect the actual outcomes.

Free printable worksheet

Addition Rule and Disjoint Events Practice

Determine whether each pair of events is disjoint, then calculate the probability of their union. Express all final answers as simplified fractions.

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Addition Rule and Disjoint Events PracticeName:

A standard six-sided die is rolled. Event A is rolling a 2. Event B is rolling a 5. Are the events disjoint? Find P(A or B).
A standard six-sided die is rolled. Event A is rolling an even number. Event B is rolling a number greater than 4. Are the events disjoint? Find P(A or B).
A card is drawn from a standard deck of 52 cards. Event A is drawing a King. Event B is drawing a Queen. Are the events disjoint? Find P(A or B).
A card is drawn from a standard deck of 52 cards. Event A is drawing a Heart. Event B is drawing an Ace. Are the events disjoint? Find P(A or B).
A ticket is selected at random from tickets numbered 1 to 20. Event A is drawing a multiple of 4. Event B is drawing an odd number. Are the events disjoint? Find P(A or B).
In a group of 30 students, 18 take Spanish, 12 take Art, and 6 take both. Find the probability that a randomly chosen student takes Spanish or Art.

Common questions

What is the difference between disjoint and mutually exclusive events?

There is no difference in probability. Disjoint and mutually exclusive are interchangeable terms that describe events that cannot occur at the same time and share zero common outcomes.

How can I tell if two events are disjoint?

List the outcomes that satisfy each event. If there is at least one outcome that satisfies both events at once, they are not disjoint. If their intersection contains zero outcomes, they are disjoint.

Can two events be both disjoint and independent?

Only if the probability of at least one of the events is zero. For events with non-zero probabilities, disjoint events cannot be independent, because if one happens, the probability of the other happening immediately drops to zero.

What happens if I forget to subtract the intersection?

If you forget to subtract the intersection, you count shared outcomes twice. Your calculated probability will be too high, and in problems with large intersections, the result may erroneously exceed 1.

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