Independent in pairs, but not all together

Question

Flip two fair coins. Define three events:

$A$ : the first coin is heads.
$B$ : the second coin is heads.
$C$ : the two coins match (both heads or both tails).

Show that $A$ , $B$ , $C$ are independent in every pair but not mutually independent.

Accepted Answer

Each event has probability 	frac12: for C, the matching outcomes are HH and TT, which is 2 of 4. Pairwise checks. Over the equally likely outcomes HH, HT, TH, TT: P(A\cap B) = P(	ext{HH}) = 	frac14 = 	frac12\cdot	frac12, P(A\cap C) = P(	ext{first H and match}) = P(	ext{HH}) = 	frac14 = 	frac12\cdot	frac12, P(B\cap C) = P(	ext{second H and match}) = P(	ext{HH}) = 	frac14 = 	frac12\cdot	frac12. All three pairs satisfy independence. The triple. For mutual independence we would also need P(A\cap B\cap C) = 	frac12\cdot	frac12\cdot	frac12 = 	frac18. But if A and B both hold (bot…

Independent in pairs, but not all together

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