The addition rule, and mutually exclusive and independent events

40 min

"Or" and "and"

A school sports teacher wants the chance that a pupil chosen at random plays football or basketball; a farmer wants the chance that all three of the seeds she plants in one hole germinate. The first is about the union of events, "or"; the second about their intersection, "and". Two rules handle them: the addition rule, and, for events that do not influence each other, the multiplication rule.

Important

Addition rule. For any two events $A$ and $B$ ,

P(A \cup B) = P(A) + P(B) - P(A \cap B) .

Reason. $A \cup B$ is made of the three non-overlapping pieces $A \cap B'$ , $A \cap B$ , $A' \cap B$ . Adding $P (A)$ and $P (B)$ counts the middle piece twice, so it is subtracted once. For three events,

P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C) .

Definition

Events $A$ and $B$ are mutually exclusive (incompatible) if they cannot occur together: $A \cap B = \varnothing$ . Then $P(A \cup B) = P(A) + P(B)$ . Events that can occur together are called inclusive.

Example: on one die, "even" and "odd" are mutually exclusive; "even" and "greater than 4" are inclusive, since 6 is both.

Worked example 1: football or basketball

In a class of 40, 22 pupils play football, 15 play basketball and 8 play both (illustrative). A pupil is chosen at random. Find the probability that the pupil plays football or basketball, and that the pupil plays neither.

Solution. $P(F \cup B) = \frac{22}{40} + \frac{15}{40} - \frac{8}{40} = \frac{29}{40}$ . Then $P(\text{neither}) = 1 - \frac{29}{40} = \frac{11}{40}$ , the 11 pupils outside both circles.

Definition

Events $A$ and $B$ are independent if the occurrence of one does not change the probability of the other; formally,

P(A \cap B) = P(A) \times P(B) .

Events $A_1, \dots, A_n$ are (mutually) independent if this product rule holds for every selection of them; then $P(A_1 \cap \dots \cap A_n) = P(A_1) \times \dots \times P(A_n)$ . Separate tosses of a coin, rolls of a die, or draws with replacement are independent.

Important

Mutually exclusive is not independent. If $P (A) > 0$ , $P (B) > 0$ and $A \cap B = \varnothing$ , then $P(A \cap B) = 0 \ne P(A)P(B)$ : knowing that $A$ happened tells you for certain that $B$ did not.

Complements of independent events are independent. If $A$ and $B$ are independent, so are $A$ and $B^{'}$ : $P(A \cap B') = P(A) - P(A \cap B) = P(A) - P(A)P(B) = P(A)\,(1 - P(B)) = P(A)P(B')$ . The same argument works for $A^{'}$ and $B^{'}$ .

Worked example 2: a surprising pair of independent events

One fair die is rolled. $A$ = "the score is even", $B$ = "the score is greater than 4". Are $A$ and $B$ independent?

Solution. $P(A) = \frac{3}{6} = \frac{1}{2}$ , $P(B) = \frac{2}{6} = \frac{1}{3}$ , and $A \cap B = \{6\}$ , so $P(A \cap B) = \frac{1}{6}$ . Since $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ , the events are independent, although they come from one roll. Among the scores 5 and 6, exactly half are even, the same proportion as among all six scores.

Worked example 3: three seeds in one hole

Each bean seed of a certain batch germinates with probability 0.8, independently of the others (illustrative). A farmer plants 3 seeds in a hole. Find the probability that all three germinate, that none does, and that at least one does.

Solution. All three: ${0.8}^{3} = 0.512$ . None: ${0.2}^{3} = 0.008$ . At least one: $1 - 0.008 = 0.992$ . Planting three seeds makes an empty hole very unlikely.

Worked example 4: testing for independence from numbers

$P (A) = 0.5$ , $P (B) = 0.4$ and $P(A \cup B) = 0.7$ . Are $A$ and $B$ mutually exclusive? Independent?

Solution. By the addition rule, $P(A \cap B) = 0.5 + 0.4 - 0.7 = 0.2$ . This is not 0, so they are not mutually exclusive. It equals $0.5 \times 0.4 = 0.2$ , so they are independent.

Warning

Add only for mutually exclusive events; multiply only for independent events. Otherwise use the full addition rule, or the conditional probabilities of lesson 3.
Do not confuse the two ideas. Mutually exclusive is about sets (no overlap); independence is about probabilities (the product rule).
"At least one" of several independent events: $1 - P(\text{none})$ , not the sum of the probabilities (which can exceed 1).

Tip

Test independence with numbers, never by intuition: compute $P(A \cap B)$ and compare it with $P (A) P (B)$ .

The Organic Chemistry Tutor: Multiplication & Addition Rule, Mutually Exclusive & Independent Events (about 10 min). Worked examples of when to add and when to multiply.

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Practice

Questions 1 to 3 are routine; 4 to 6 need more care; 7 asks for a reason.

$P (A) = 0.6$ , $P (B) = 0.3$ , $P(A \cap B) = 0.1$ . Find $P(A \cup B)$ and $P(A' \cap B')$ .
A card is drawn from the 26 letters of the alphabet written on cards. Find the probability that it is a vowel or one of the letters of the word "RWANDA".
A fair coin is tossed 4 times. Find the probability of 4 heads, and of at least one tail.
$A$ and $B$ are independent with $P (A) = 0.3$ and $P (B) = 0.5$ . Find $P(A \cup B)$ and $P(A \cap B')$ .
Two machines work independently; on a given day each breaks down with probabilities 0.05 and 0.1 (illustrative). Find the probability that at least one breaks down.
Of 100 pupils, 60 study Biology, 50 study Chemistry and 35 study both. Are "studies Biology" and "studies Chemistry" independent for a pupil chosen at random?
Explain why two events with non-zero probabilities cannot be both mutually exclusive and independent.

Show answersHide answers

$0.6 + 0.3 - 0.1 = 0.8$ ; $P(A' \cap B') = 1 - 0.8 = 0.2$ .
Vowels A, E, I, O, U (5); letters of RWANDA: R, W, A, N, D (5); both: A. $P = \frac{5 + 5 - 1}{26} = \frac{9}{26}$ .
$\frac{1}{16}$ ; $1 - \frac{1}{16} = \frac{15}{16}$ .
$0.3 + 0.5 - 0.15 = 0.65$ ; $P(A \cap B') = 0.3 \times 0.5 = 0.15$ .
$1 - 0.95 \times 0.9 = 1 - 0.855 = 0.145$ .
$P(B)P(C) = 0.6 \times 0.5 = 0.3 \ne 0.35 = P(B \cap C)$ : not independent.
Mutually exclusive gives $P(A \cap B) = 0$ ; independent gives $P(A \cap B) = P(A)P(B) > 0$ . Both cannot hold.

Sources and credits

Written by Ganzaa: the explanations, the Venn diagram, the worked examples, the practice items and the quiz questions. The class, seed and machine figures are illustrative. Every probability was checked by computer.
Video: Multiplication & Addition Rule - Probability - Mutually Exclusive & Independent Events, The Organic Chemistry Tutor, embedded from YouTube.

1"Or" and "and"

"Or" and "and"