Let the sample space S consist of the 3! permutations of letters a, b and c along with three triples of each letter (e. g. aaa) and let each element of S have probability 1/9. Define Ai = {i-th place in thee triple is occupied by a}, i = 1, 2, 3. Verify whether A1, A2 and A3 are independent?

a)P(A1) = P( aaa, abc, acb ) = 1/3

    P(A2) = P(aaa, bac, cab ) = 1/3

      P(A3) = P(aaa, cba, bca ) = 1/3

b)  To verify IF A1, A2 and A3 are independent  we can see that

P( A1 and A2 and A3 ) = P( aaa ) = 1/9  is not equal to P(A1)P(A2)P(A3)

Step-by-step explanation:

Based on the given conditions The sample space S can be written as

aaa, bbb, ccc, abc, acb, bac, bca, cab,and cba  = 9 most likely outcomes

Defining Ai ;which is the event that ith place in the triple is occupied by a

The probabilities of i=1,2,3 can be computed as

P(A1) = P( aaa, abc, acb ) = 1/3

P(A2) = P(aaa, bac, cab ) = 1/3

P(A3) = P(aaa, cba, bca ) = 1/3

P(A1 and A3) = P(A2 and A3 ) = P ( A1 and A3 ) =  1/9

THIS CAN BE EXPRESSED AS

P(A1)P(A2) = P(A2)P(A3) = P(A1)P(A3)

It can observed that

A1, A2, A3 are pair wise independent of each other

to verify this we can see that

P( A1 and A2 and A3 ) = P( aaa ) = 1/9  is not equal to P(A1)P(A2)P(A3)