A First Course in Quality Engineering, 2nd Edition

Statistics for Quality 79

probability that follows makes use of conditional probabilities and gives a

very useful result when only conditional information is available�

2.5.1.3.8  The Theorem of Total Probability

Let B

1

, B

2

,…, B

k

be partitions of a sample space S such that (B

1

∪ B

2

∪ … ∪ B

k

) = S 

and (B

i

∩ B

j

) = ∅ (null set) for any pair i and j (see diagram below)� The partitions

are mutually exclusive events that jointly make up the sample space� If A is an

event of interest in the same sample space, then:

P A P A B P B P A B P B P A B P B

k k

( )

=

( )

( )

+

( )

( )

+ +

( )

( )

| | |

1 1 2 2



B

1

B

2

B

3

B

6

B

5

B

4

A

S

The individual product terms on the right ...