Question

We toss a fair coin twice, define A ={the first toss is head}, B ={the second toss is tail}, and C={the two tosses have different outcomes}. Show that events A, B, and C are pairwise independent but not mutually independent.

Answer 1

We are given here that:
A ={the first toss is head},
B ={the second toss is tail},
C={the two tosses have different outcomes}

P( A and B) = P( first coin id heads and second toss is tail) = 0.5*0.5 = 0.25

P(A) = P(B) = 0.5
Therefore, P(A and B) = P(A)P(B) = 0.25

Therefore A, B are pairwise independent events.

P(C) = P(HT) + P(TH) = 0.25 + 0.25 = 0.5

P(A and C) = P(first coin toss is head and second is tail) = 0.25 = P(A)P(C)
Therefore A and C are pairwise independent.

P(B and C) = P(second coin is tail and first is heads) = 0.25 = P(B)P(C)
Therefore B and C are pairwise independent events.

P(A and B and C) = P(first toss is heads and second is tail) = 0.25 which is not equal to P(A)P(B)P(C) = 0.125

Therefore: A, B and C are not mutually independent.