Page 37 - Probability and Statistical Inference

14 1. Nations of Probability

1.4.2 Bayes’s Theorem

We now address another type of situation highlighted by the following
example.
Example 1.4.5 Suppose in another room, an experimenter has two urns at
his disposal, urn #1 and urn #2. The urn #1 has eight green and twelve blue
marbles whereas the urn #2 has ten green and eight blue marbles, all of same
size and weight. The experimenter selects one of the urns at random with equal
probability and from the selected urn picks a marble at random. It is announced
that the selected marble in the other room turned out blue. What is the prob-
ability that the blue marble was chosen from the urn #2? We will answer this
question shortly. !
The following theorem will be helpful in answering questions such as the
one raised in the Example 1.4.5.
Theorem 1.4.3 (Bayes’s Theorem) Suppose that the events {A , ..., A }
k
1
form a partition of the sample space S and B is another event. Then,

Proof Since {A , ..., A } form a partition of S, in view of the Theorem 1.3.1,
1
k
part (vi) we can immediately write

by using (1.4.4). Next, using (1.4.4) once more, let us write

The result follows by combining (1.4.6) and (1.4.7). !
This marvelous result and the ideas originated from the works of Rev. Tho-
mas Bayes (1783). In the statement of the Theorem 1.4.3, note that the condi-
tioning events on the rhs are A , ..., A , but on the Ihs one has the conditioning
1
k
event B instead. The quantities such as P(A ), i = 1, ..., k are often referred to as
i
the apriori or prior probabilities, whereas P(A | B) is referred to as the poste-
j
rior probability. In Chapter 10, we will have more opportunities to elaborate
the related concepts.
Example 1.4.6 (Example 1.4.5 Continued) Define the events
A : The urn #i is selected, i = 1, 2
i
B: The marble picked from the selected urn is blue