what is the minimum number of points a sample space must contain

Question

1. what is the minimum number of points a sample space must contain in order that there exist n independent events A1, A2,...An，none of which has probability of 0 or 1?

Answer 1

Assume minimum number of sample points as 0 i.e P(A)=0 and P(B)=0 this implies P(A and B)=0 but then A and B become disjoint and thus by above proof they cannot be independent Assume the minimum number of sample points to be 1 i.e P(A)=1 and P(B)=0 implies P(A and B)=0 i.e A and B are disjoint |||ly P(A)=0 and P(B)=1 cannot be mutually independent

let P(A)=1 and P(B)=1 i.e they refer to same sample point then A and B refer to the same event and hence cannot be mutually independent Assume the minimum number of sample points to be 2 e.g let us toss a biased coin(coin that always shows a head) and an unbiased coin one after the other. thus sample points are HH and HT let A refer to showing a head on biased coin let B refer to showing a head on unbiased one P(A)=1 and P(B)=1/2 P(A and B)=1/2 = P(A)*P(B) and thus they are independent.

thus minimum number of points in a sample space if there are two events A and B that are independent of each other is 2.