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Statistics

Notes Since the coins are given to be unbiased, the elementary events are equally likely, therefore

1 3 1 1
( ) 
( ) 
P A , P ( ) B  , P C , P (A  ) B  P (A C  )
4 4 2 4
(a) We have to determine P (A/B)
P (A  ) B 1 4 1
)
P ( /A B   ´ 
P ( ) B 4 3 3
(b) We have to determine P(A/C)

P (A  C ) 1 2 1
)
P ( /A C   ´ 
( )
P C 4 1 2
7.3 Theorems on Probability

Theorem 6. Bayes Theorem or Inverse Probability Rule

The probabilities assigned to various events on the basis of the conditions of the experiment or
by actual experimentation or past experience or on the basis of personal judgement are called
prior probabilities. One may like to revise these probabilities in the light of certain additional
or new information. This can be done with the help of Bayes Theorem, which is based on the
concept of conditional probability. The revised probabilities, thus obtained, are known as posterior
or inverse probabilities. Using this theorem it is possible to revise various business decisions in
the light of additional information.

Bayes Theorem

If an event D can occur only in combination with any of the n mutually exclusive and exhaustive
events A , A , ...... A and if, in an actual observation, D is found to have occurred, then the
1 2 n
probability that it was preceded by a particular event A is given by
k
P ( ) ( /A .P D A )
P (A k /D )  n k k
å P ( ) ( /A i .P D A i )
i 1
Proof.
Since A , A , ...... A are n exhaustive events, therefore,
n
2
1
S  A  A ......  A .
n
1
2
Since D is another event that can occur in combination with any of the mutually exclusive and
exhaustive events A , A , ...... A , we can write
1
n
2
D  b A  g D  b A  Dg
D b
1 A  g ...... n
2
Taking probability of both sides, we get
P D b g b 1 Dg b 2 Dg + ...... + P A b n  Dg
P A 
P A 
+

We note that the events A  b A  Dg , etc. are mutually exclusive.
1 b
Dg,
2
n n
P D ( P A  D )  å P ( ) ( /A P D A i ) .... (1)
.
( )  å
i
i
i 1 i 1 
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