Question

A coin is tossed three times. Let the events A, B and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.
Check the independence of (i) A and B (ii) B and C and (iii) C and A

Answer 1

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