An urn contains four tickets with numbers `112`, `1121`, `211`, `222` and one ticket is drawn. Let `A_i (i=1,2,3)` be the event that the `i^(th)` digit of the number on ticket drawn is 1. Discuss the independence of the events `A_1`, `A_2`, `A_3`

Explanation

We have `P(A_1)=2/4=1/2= P(A_2)= P(A_3)`
`A_1∩A_2`is the event that the first two digits in the numbers drawn are each equal to 1 and
so `P(A_1∩A_2)=1/4=1/2*1/2=P(A_1)P(A_2)`
Similarly, `P(A_2∩A_3)=P(A_2)P(A_3)` and `P(A_3∩A_1)=P(A_3)P(A_1)`.
Thus the events `A_1,A_2 and A_3` are equal to 1.
and since there is no such number,
we have
`P(A_1∩A_2∩A_3)=0= P(A_1)P(A_2)P(A_3)`
Hence the events `A_1,A_2,A_3` are not mutually independent although they are pairwise independent.