2.5.1 The Multiplication Rule for Intersections

• Frequently the nature of an experiment suggests that
  • two events $A$ and $B$ should be assumed independent.
• This is the case, for example,
  • if a manufacturer receives a circuit board from each of two different suppliers,
  • each board is tested on arrival,
  • $A=\{$ first is defective $\}$
  • $B=\{$ second is defective $\}$ .
• If $P\left(A\right)=.1$ ,
  • it should also be the case that $P\left(A\mid B\right)=.1$ ;
  • knowing the condition of the second board shouldn’t provide information about the condition of the first.
• The probability that both events will occur is easily calculated from the individual event probabilities
  • when the events are independent.

independence

$A$ and $B$ are independent if and only if (iff)

$$P\left(A\cap B\right)=P\left(A\right)\cdot P\left(B\right)\text{\hspace{1em}(2.8)}$$

The verification of this multiplication rule is as follows:

$$\begin{array}{rl}P\left(A\cap B\right)& =P\left(A\mid B\right)\cdot P\left(B\right)\\ & =P\left(A\right)\cdot P\left(B\right)\end{array}\text{\hspace{1em}(2.9)}$$

where the second equality in Equation (2.9) is valid

• iff $A$ and $B$ are independent. Equivalence of independence and Equation (2.8) imply that
• the latter can be used as a definition of independence.

EX 2.34 washer vs dryer

EX 2.35 batch pass inspection