Page 53 - DMTH404_STATISTICS

Unit 3: Conditional Probability and Independence Baye’s Theorem

 We have stated and proved Bayes’ theorem : Notes
If B , B , . . . , B , are n events which constitute a partition of , and A is an event of positive
1 2 n
probability, then
P(B )P(A|B )
P(B |A)  n r r
r
 P(B )P(AIB )
j
j
1
for any r, 1  r  n.
 We have defined and discussed the consequences of independence of two or more events.
 We have seen the method of assignment of probabilities when dealing with independent
repetitions of an experiment.

3.6 Keywords

Conditional probability: The concept of conditional probability P(A | B) of a given the event B.

P(A  B)
P(A|B) = , P(B) 0

P(B)
Bayes’ theorem: If B , B , . . . , B , are n events which constitute a partition of W, and A is an event
1 2 n
of positive probability, then
P(B )P(A|B )
P(B |A)  n r r
r
 P(B )P(AIB )
j
j
1
for any r, 1  r  n.
3.7 Self Assesment

1. There are 1000 people. There 400 females and 200 colour-blind person. Find the probability
of females.

(a) 0.4 (b) 0.6
(c) 0.16 (d) 0.21

 n 
2. There are   points with j successes and (n – j) failures. Since each such point carries the
j
 
n-j
i
probability p q , the probability of j successes, denoted by b(j, n, p) is
 n 

j
b(j, n, p) =   p q n j , 0, 1 , ....n.
j
 
(a) Binomial probabilities (b) Conditional probability
(c) Bayer’s theorem (d) Clanical probability
These are called binomial probabilities.
3. the concept of conditional probability P(A | B) of a given the event B.

P(A  B)
P(A|B) = , P(B) 0

P(B)

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