Mutually exclusive events and the addition rule

Why this exists

Basic probability finds the chance of an event by counting its outcomes. Often, though, an event is described in pieces: “red or blue”, “a sum of 5 or two even numbers”, “the train is late or canceled”. Each piece may already have a known chance, and recounting from scratch every time wastes that work. Worse, sometimes there is nothing to count: a chance like 0.13 for “the train is early” comes from records, not from a list of equally likely outcomes.

So the question is when the chance of “A or B” can be built straight from the chance of A and the chance of B. The answer is simple when the two events cannot happen together: add them. This lesson says exactly when that is allowed, why it works, and why “cannot happen together” is a different idea from “has no effect on each other”.

Definition

Two events A and B on the same trial are mutually exclusive when they cannot both happen on that trial. In the language of sample spaces and events: no outcome belongs to both of them, so their intersection is the empty set,

Mutually exclusive events are also called disjoint, the word used for sets with no member in common in Union, intersection and complement.

Several events are mutually exclusive when every pair of them is: no outcome lies in two of them at once.

The event “A or B” happens when at least one of A and B happens. In probability, “or” always includes the case where both happen, unless the problem says otherwise. Its probability is written P(A \text{ or } B), or P(A \cup B) with the union sign. For mutually exclusive events the “both” case never arises, so the two readings of “or” give the same event.

Intuition

A bag holds 20 marbles: 7 red, 5 blue and 8 green. One marble is drawn without looking, so each of the 20 is equally likely. A marble has one color, so “red” and “blue” cannot both happen: they are mutually exclusive. Figure 1 shows them as two circles that do not touch.

The marbles that are red or blue are the ones in either circle. Since no marble is in both, the count is just 7 + 5 = 12, and

The same answer comes from the two separate chances, because dividing a sum by 20 is the same as dividing each part by 20:

Nothing was lost and nothing was counted twice. Figure 2 shows when that is guaranteed. On the left the circles are apart, and the two shaded areas together cover each outcome once. On the right the circles overlap; adding the two areas covers the overlap twice, and the sum is too big.

Results

Worked examples

Boundaries

The rule needs the events to be mutually exclusive. For events that can happen together, P(A) + P(B) counts the outcomes in both twice and is too big, as Example 2 showed. The rule that works for every pair of events is the general addition rule.

For three or more events, every pair must be checked. It is not enough for A to miss B and B to miss C; if A and C share an outcome, adding all three chances counts it twice.

The events belong to one trial. “Mutually exclusive” asks whether a single outcome can lie in both events. Events about two separate stages, like drawing a tile and then rolling a solid, usually can happen together.

Whether events overlap is read from the outcomes, not from the words. “Rains today” and “is sunny today” overlap on a day with sun and showers, unless the model says each day is exactly one kind. The model, or the list of outcomes, decides.

A probability of 0 is a weaker statement than “cannot happen”. When every outcome has a chance above 0, as in all the examples here, P(A \text{ and } B) = 0 means A and B share no outcome. In some models an outcome itself has chance 0 (a point picked at random on a line segment lands exactly at its middle with probability 0, yet it can land there), and then P(A \text{ and } B) = 0 does not prove the events are disjoint. The addition rule still holds in that case.

Mutually exclusive and independent together happens only when one of the events has probability 0; otherwise the two ideas exclude each other, as the Results say.

Where this goes next

The addition rule is half of the toolkit for compound events; the multiplication rule for independent events and for Dependent events and drawing without replacement is the other half. They meet in tree diagrams: the paths through a tree are mutually exclusive, so the chances of several paths add.

Events that overlap need the general addition rule, which takes away the doubly counted part, and the language of unions and intersections of events for writing it down. Splitting a whole population into groups that cannot overlap, then adding each group’s share of an event, is the law of total probability, built on Conditional probability. And the values of a random variable are mutually exclusive outcomes, which is why the probabilities in a probability distribution add up to 1.

Exercises

Mutually exclusive or not

1. A number cube is rolled once. For each pair, say whether the events are mutually exclusive, and give the outcomes that settle it. (a) “a 1” and “an even number”; (b) “a multiple of 3” and “a 4 or a 5”; (c) “an odd number” and “a prime number”; (d) “a 6” and “more than 4”.
2. A bag holds 26 tiles, one for each letter A to Z, and one tile is drawn at random. The vowels are A, E, I, O and U. (a) Are “a vowel” and “a letter of the word RHYTHM” mutually exclusive? If so, find the probability of one or the other. (b) Are “a vowel” and “a letter of the word PLANET” mutually exclusive?

Using the addition rule

3. A box holds 30 counters: 9 yellow, 12 white, 4 black and 5 orange. One is drawn at random. Find (a) P(\text{yellow or black}), (b) P(\text{not white}) in two ways, (c) P(\text{white, black or orange}).
4. A number cube is rolled. Find the probability of a 2 or a number more than 4.
5. A ferry ticket chosen at random is a child ticket with probability 0.2, an adult ticket with probability 0.55 and a senior ticket with probability 0.25, and every ticket is exactly one of these. Find the probability that it is a child or senior ticket, by adding and by using the complement.
6. Two number cubes are rolled. Find the probability of a sum of 3 or a sum of 12.
7. Two number cubes are rolled. Find the probability that the sum is a multiple of 5 by splitting it into mutually exclusive pieces.
8. The twelve-faced solid of Example 2 is rolled. Find the probability of a prime number or a multiple of 4, after checking that the rule applies.

Finding a missing probability

9. Events A and B are mutually exclusive, P(A) = 0.42 and P(A \text{ or } B) = 0.7. Find P(B).
10. Every egg from a farm is graded as exactly one of small, medium or large. For an egg chosen at random, P(\text{small}) = 0.18 and P(\text{large}) = 0.35. Find P(\text{medium}) and P(\text{small or medium}).
11. Events A, B and C are mutually exclusive and between them hold every outcome. P(A) = 2P(B) and P(C) = 0.4. Find P(A) and P(B).
12. Can two mutually exclusive events have P(A) = 0.6 and P(B) = 0.5? Explain with the addition rule.

Mutually exclusive or independent

13. P(A) = 0.2, P(B) = 0.5 and P(A \text{ and } B) = 0.1. Are A and B mutually exclusive? Are they independent?
14. A and B are mutually exclusive, with P(A) = 0.25 and P(B) = 0.4. Find P(A \text{ and } B) and P(A \text{ or } B), and decide whether they are independent.
15. A tile is drawn from the letter bag of Exercise 2, and then the twelve-faced solid of Example 2 is rolled. A: a vowel. B: the solid shows 12. Are A and B mutually exclusive? Are they independent? Find P(A \text{ and } B).
16. A number cube is rolled once. A: “a 1 or a 2”. B: “a 5 or a 6”. Find P(A \text{ or } B). Then find P(B), and the chance of B once you know A happened, and say whether A and B are independent.

Explaining

17. Thorvald rolls a number cube and writes \displaystyle P(\text{odd}) + P(\text{less than } 4) = \frac{1}{2} + \frac{1}{2} = 1, so the roll is certain to be odd or less than 4. Explain what went wrong and find the correct probability.
18. Prove that if A and B are mutually exclusive, P(A) > 0 and P(B) > 0, then A and B are not independent.
19. Use the addition rule to show that P(\text{not } A) = 1 - P(A) for any event A.
20. On one roll of a number cube, A = \{1, 2\}, B = \{3, 4\} and C = \{4, 5, 6\}. Adding P(A) + P(B) + P(C) gives more than 1. Explain which condition of the rule fails, and find P(A \text{ or } B \text{ or } C).
21. In a sample space of equally likely outcomes, A and B are mutually exclusive and P(A) + P(B) = 1. Explain why every outcome lies in exactly one of A and B.