Question

A coin is tossed three times. If events A and B are defined as A = Two heads come, B = Last should be head. Then, A and B are
(a) independent
(b) dependent
(c) both
(d) mutually exclusive

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)\text{dependent} \\ S=\left[\left(HHH\right),\left(HHT\right),\left(HTH\right),\left(HTT\right),\left(THH\right),\left(THT\right),\left(TTH\right),\left(TTT\right)\right] \\ P\left(A\right)=P\left(2\text{heads}\right)=\frac{3}{8} \\ P\left(B\right)=P\left(\text{last one is head}\right)=\frac{4}{8} \\ P\left(A\cap B\right)=\frac{2}{8}=\frac{1}{4}\ne P\left(A\right)P\left(B\right) \\ \mathrm{Thus},A\mathrm{and}B\mathrm{are}\text{dependent.} \end{gathered}$$