Combinatorics • Topic 22

Independent Events

Independence is a precise mathematical concept: knowing that one event occurred gives no information about whether the other occurred.

Definition

Definition (Independence)

Two events

A

and

B

are independent if and only if:

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

Equivalently (if $P (B) > 0$ ):

P (A ∣ B) = P (A)

"The probability of

A

given

B

is just the probability of

A

These are often confused but are opposites!

Mutually Exclusive (Disjoint): If $A$ happens, $B$ cannot happen. ( $A \cap B = \emptyset$ ). They are highly dependent.
Independent: If $A$ happens, the chance of $B$ happening is unchanged.

For 3 events

A, B, C

Pairwise Independent: $P (A \cap B) = P (A) P (B)$ , etc., for all pairs.
Mutually Independent: Also requires $P (A \cap B \cap C) = P (A) P (B) P (C)$ .

Example

Roll a red die and a blue die.

A

: Red is 1.

B

: Blue is 1.

C

: Sum is 7. Are

A

and

B

independent? Yes. Are

A

and

C

independent?

P (A) = 1/6

P (C) = 6/36 = 1/6

P (A \cap C)

: Red is 1 and Sum is 7

⟹

(1,6). Prob is

1/36

P (A) P (C) = 1/36

. Yes! Knowing Red is 1 doesn't change the odds of Sum 7.

Exercise (Problem 1)

P (A) = 0.4

P (B) = 0.5

, and

A, B

are independent, find

P (A \cup B)

Exercise (Problem 2)

Prove that if

A

and

B

are independent, then

A^{c}

and

B

are also independent.

Exercise (Problem 3)

A system fails if components A AND B both fail. A fails with prob 0.1, B with prob 0.2 (independently). What is the probability the system works?