Let A_1,A_2,...,A_n be pairwise independent events. Are…

Let $A_1,A_2,\dots,A_n$ be pairwise independent events. Are these events always independent as a whole (mutually independent), i.e. is the conditional probability of any one of them, given that any other events of this set have occurred, always equal to its unconditional probability?

Step 1: The definitions

• Pairwise independent: $P(A_iA_j)=P(A_i)P(A_j)$ for all $i\ne j$.
• Mutually independent: $P(A_{i_1}\cdots A_{i_k})=P(A_{i_1})\cdots P(A_{i_k})$ for every choice of different indices. Equivalently, the conditional probability of any $A_i$ given any combination of the others equals $P(A_i)$.

Step 2: Bernstein's example

Toss a fair coin twice. Let

\[ A_1=\text{"first toss heads"},\quad A_2=\text{"second toss heads"},\quad A_3=\text{"both tosses show the same side"}. \]

Each outcome has probability $\frac14$, and each event has probability $\frac12$.

Step 3: Pairwise independent

$A_1A_2=A_1A_3=A_2A_3=\{\text{HH}\}$, each of probability $\frac14=\frac12\cdot\frac12$. ✓

Step 4: Not mutually independent

\[ P(A_1A_2A_3)=P(\{\text{HH}\})=\frac14\ne\frac18=P(A_1)P(A_2)P(A_3). \]

In the language of conditional probability: $P(A_3\mid A_1A_2)=1\ne\frac12=P(A_3)$. Knowing that both tosses gave heads tells us for sure that they are the same.

Step 5: The answer

\[ \boxed{\text{No: pairwise independence does not imply independence as a whole.}} \]

(Conversely, mutual independence always implies pairwise independence.)

Step 6: Another example — a tetrahedron

A fair tetrahedron has one red face, one blue face, one green face and one face with all three colours. Let $R$, $B$, $G$ = "the bottom face contains red / blue / green". Each has probability $\frac12$, each pair $\frac14$, but all three occur together with probability $\frac14\ne\frac18$.