Mutually exclusive and independent events

The two key relationships between events: mutually exclusive (they cannot both happen, so P(A∩B) = 0 and P(A∪B) = P(A) + P(B)) and independent (one does not affect the other, so P(A∩B) = P(A)×P(B)). How to use each rule, how to TEST whether two events are independent, and why mutually exclusive and independent are not the same thing.

Statistics · Probability

Mutually exclusive and independent events

Why it works

Two events can be related in two very different ways, each with its own rule.

Mutually exclusive events cannot both happen at once — rolling a

2

and rolling a

5

on one die, say. Their circles on a Venn diagram do not overlap, so

P(A \cap B) = 0, \qquad \text{and therefore} \qquad P(A \cup B) = P(A) + P(B).

The addition formula loses its

-P(A\cap B)

term because there is no overlap to double-count.

Independent events are ones where one happening does not change the probability of the other — two separate coin flips, or drawing a card, replacing it, and drawing again. For independent events,

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

Testing for independence. You are often given

P(A)

P(B)

and

P(A\cap B)

and asked whether

A

and

B

are independent. Compare:

if $P(A\cap B) = P(A)\times P(B)$ , the events are independent;
if $P(A\cap B) \ne P(A)\times P(B)$ , they are not independent.

So work out

P(A)\times P(B)

and check it against the actual

P(A\cap B)

.

They are not the same idea. Mutually exclusive is about whether they can co-occur; independent is about whether they influence each other. In fact, if

A

and

B

both have non-zero probability and are mutually exclusive, they are not independent: if

A

happens then

B

certainly cannot, so

A

has changed

B

's probability (to

0

). Don't reach for

P(A)+P(B)

unless the events are mutually exclusive, and don't reach for

P(A)\times P(B)

unless they are independent.