STAT 3401, Intro. Prob. Theory 18/26 Jaimie Kwon
1/24/2005
2.10 The law of total probability and Bayes’ rule
♦ Definition 2.11. For some positive integer k, let the sets B1,B2,…,Bk be such that
1. S= B1∪B2∪…∪Bk ,
2. Bi∩Bj=∅ if i≠j
Then the collection of sets {B1,B2,…,Bk} is said to be a “partition” of S.
♦ If A is any subset of S, and {B1,B2,…,Bk} is a partition of S, A can be “decomposed” as:
A=(A∩B1)∪(A∩B2) ∪…∪ (A∩Bk)}
* See the Venn Diagram
♦
Theorem 2.8. (The law of total probability) Assume that {B
1
,B
2
,…,B
k
} is a partition of S such that
P(B
i
)>0 for i=1,…,k. Then for any event A
P(A)=Σ
i=1,…,k
P(A|B
i
)P(B
i
)
à Proof: apply the additive law
à Sometimes, it’s easier to calculate P(A|Bi) for a suitably chosen partition than to compute
P(A) directly.
♦
Theorem 2.9. (Bayes’ Rule) Assume {B
1
,B
2
,…,B
k
} is a partition of S such that P(B
i
)>0 for
i=1,2,…,k. Then
∑
=
=k
i
ii
jj
j
BPBAP
BPBAP
ABP
1
)()|(
)()|(
)|(
.
♦ Example 2.23. (An electronic fuse) 5 production lines produce fuses at the same production
rate, with 2% defect rate except line 1 with 5% defect rate. A customer tested three fuses and
one of them failed. What is:
P(the lot was produced in line 1|the data)=?
P(the lot was produced in one of lines 2-4|the data)=?
♦ HW. Some of the exercises 2.98~116
♦ Keywords: the law of total probability; Bayes rule
