2.5 Independence

• The definition of conditional probability enables us to revise the probability $P\left(A\right)$ originally assigned to $A$ when we are subsequently informed that another event $B$ has occurred;
  • the new probability of $A$ is $P\left(A\mid B\right)$ .
• In our examples, it frequently happened that $P\left(A\mid B\right)$ differed from the unconditional probability $P\left(A\right)$ .
• Then the information ”$B$ has occurred” resulted in a change in the likelihood of $A$ occurring.
• Often the chance that $A$ will occur or has occurred is not affected by knowledge that $B$ has occurred, so that
• $P\left(A\mid B\right)=P\left(A\right)$
• It is then natural to regard $A$ and $B$ as independent events, meaning that the occurrence or nonoccurrence of one event has no bearing on the chance that the other will occur.

independent vs dependent

Two events $A$ and $B$

• are independent if $P\left(A\mid B\right)=P\left(A\right)$
• are dependent otherwise.

• The definition of independence might seem “unsymmetric”
  • because we do not also demand that
  • $P\left(B\mid A\right)=P\left(B\right)$
  • However, using the definition of conditional probability and the multiplication rule,

$$P\left(B\mid A\right)=\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{P\left(A\mid B\right)P\left(B\right)}{P\left(A\right)}\text{\hspace{1em}(2.7)}$$

• The right-hand side of Equation (2.7) is $P\left(B\right)$
  • if and only if $P\left(A\mid B\right)=P\left(A\right)$ (independence).
  • Thus the equality in the definition implies the other equality (and vice versa).
• It is also straightforward to show that
  • if $A$ and $B$ are independent,
  • then so are the following pairs of events:
    1. $A^{\prime }$ and $B$
    2. $A$ and $B^{\prime }$ , and
    3. $A^{\prime }$ and $B^{\prime }$ .

EX 2.32 pumps in gas stations

EX 2.33 mutually exclusive vs independent

2.5.1 The Multiplication Rule for Intersections

2.5.2 Independence of More Than Two Events

EXERCISES Section 2.5 (70-89)