Question

A fair coin and an unbiased die are tossed. LEt A be he event 'head appears on the coin' and B be the events '3 be the events '3 on the die', Cheek whether A and B are independent events or not.

Answer 1

If a fair coin and an unbiased die are tossed , then the sample space S is given by, S = $$\left\{\begin{array}{cccccc}(H,1),& (H,2),& (H,3),& (H,4),& (H,5),& (H,6)\\ (T,1),& (T,2),& (T,3),& (T,4),& (T,5),& (T,6)\end{array}\right\}$$ $\Rightarrow n\left(S\right)=12$
Also, A : head appears on the coin B:3 appears on the die
$\therefore A=$ set of events having head on one coin
$=\left\{\right(H,1),(H,2),(H,3),(H,4),(H,5),(H,6\left)\right\},B=setofeventshaving3ononedie=\left\{\right(H,3),(T,3\left)\right\}\Rightarrow A\cap B=\left\{\right(H,3\left)\right\}\Rightarrow \Rightarrow n\left(A\right)=6,n\left(B\right)2n(A\cap B)=1Hence,P\left(A\right)=\frac{6}{12}=\frac{1}{3},P\left(B\right)=\frac{2}{12}=\frac{1}{6}=andP(A\cap B)=\frac{1}{12}Now,P\left(A\right)\times P\left(B\right)=\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}=P(A\cap B)$
Therefore, A and B are independent events.