2.5. SOME BASIC OPERATIONS OF PROBABILITY 41
2.5.8 BAYES’ RULE
Bayes’ rule (alternatively Bayes’ theorem or Bayes’ law) describes the probability of an event,
based on prior knowledge of conditions that might be related to the event.
e Bayes’ rule is an alternative expression of the conditional probability. If A and B are
two events, since P .A B/ D P .B A/, we can obtain the following equation according to
Equations (2.21) or (2.22) of the conditional probability,
P .A B/ D P .A
j
B /P .B/ D P .B A/ D P .B
j
A/P .A/:
Rearranging this equation, we can have:
P
.
B
j
A
/
D
P
.
A
j
B
/
P
.
B
/
P
.
A
/
: (2.25)
In Equation (2.25), P .B
j
A/ is the probability of event B when event A has happened. In the
Bayes’ rule, the probability P .B
j
A/ at the left end of the formula is typically called as the
posterior probability. e probability P .A
j
B / at the right end of the formula is the probability
of occurrence of the event A when the event B has occurrence and is typically called as the prior
probability.
A general expression of the Bayes’ rule can be described as follows.
If B
1
, B
2
,…,B
n
are a set of mutually exclusive, exhaustive collective event, that is,
B
j
B
i
D ;I j ¤ i; i; j D 1; 2; : : : ; n
D B
1
[ B
2
[ : : : [ B
n
the probability of
P .B
j
j
A/ will be
P
B
j
j
A
D
P
A
ˇ
ˇ
B
j
P
B
j
P
iDn
iD1
P
.
B
i
/
P
.
A
j
B
i
/
: (2.26)
Example 2.23
In an engineering statics class, 73% of students passed this class. For this class, students have a
chance for additional learning from tutoring offered by the school. For this class, 18% of students
attend the additional tutoring section. It is noted that 95% of students who have attended the
tutoring section will pass the class. Calculate the probability of a student with a passing grade
who attended the tutoring section.
Solution:
Let us use event A to represent a student with a passing grade and event T to represent attending
additional tutoring section.
From the giving information, we have:
P .T / D 0:18I P .A
j
T / D 0:95I P .A/ D 0:73:
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