Question

A pair of fair coins is tossed yielding the equiprobable space S={HH, HT, TH, TT}. Consider the events:
A={head on first coin}={HH, HT},
B={head on second coin}={HH, TH}
C={head on exactly one coin}={HT, TH}
Then check whether $A,B,C$ are independent or not.

Answer 1

$P\left(A\right)=P\left(B\right)=P\left(C\right)=\frac{2}{4}=\frac{1}{2}$
Also $P(A\cap B)=\frac{1}{4}=P\left(A\right)P\left(B\right),P(A\cap C)=\frac{1}{4}=P\left(A\right)P\left(C\right),P(B\cap C)=\frac{1}{4}=P\left(B\right)P\left(C\right)$
but $P(A\cap B\cap C)=0\ne P\left(A\right)P\left(B\right)P\left(C\right)$
$\therefore$ $A,B$ & $C$ are not independent