Question

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
(i) A = the first throw results in head, B = the last throw results in tail
(ii) A = the number of heads is odd, B = the number of tails is odd
(iii) A = the number of heads is two, B = the last throw results in head

Answer 1

$$\begin{gathered} S=\left[\left(HHH\right)\left(HHT\right)\left(HTH\right)\left(HTT\right)\left(THH\right)\left(THT\right)\left(TTH\right)\left(TTT\right)\right] \\ \left(\mathrm{i}\right)P\left(A\right)=\frac{4}{8}=\frac{1}{2} \\ P\left(B\right)=\frac{4}{8}=\frac{1}{2} \\ \mathrm{Now}, \\ P\left(A\cap B\right)=\frac{2}{8}=\frac{1}{4} \\ P\left(A\cap B\right)=P\left(A\right)P\left(B\right) \\ \text{Thus, A and B are independent events.} \\ \left(\mathrm{ii}\right)P\left(A\right)=\frac{4}{8}=\frac{1}{2} \\ P\left(B\right)=\frac{4}{8}=\frac{1}{2} \\ \mathrm{Now}, \\ P\left(A\cap B\right)=\frac{0}{8}=0 \\ P\left(A\cap B\right)\ne P\left(A\right)P\left(B\right) \\ \text{Thus, A and B are not independent events.} \\ \left(\mathrm{iii}\right)P\left(A\right)=\frac{3}{8} \\ P\left(B\right)=\frac{4}{8}=\frac{1}{2} \\ \mathrm{Now}, \\ P\left(A\cap B\right)=\frac{2}{8}=\frac{1}{4} \\ P\left(A\cap B\right)\ne P\left(A\right)P\left(B\right) \\ \text{Thus, A and B are not independent events.} \end{gathered}$$