Question

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Answer 1

$$\begin{gathered} \mathrm{Suppose}S\mathrm{represents}\mathrm{the}\mathrm{event}\mathrm{of}\mathrm{buying}\mathrm{a}\mathrm{shirt}\mathrm{and}T\mathrm{represents}\mathrm{the}\mathrm{event}\mathrm{of}\mathrm{buying}\mathrm{a}\mathrm{trouser}. \\ \mathrm{We}\mathrm{have}, \\ P\left(S\right)=0.2 \\ P\left(T\right)=0.3 \\ P\left(S/T\right)=0.4 \\ \mathrm{Now}, \\ P\left(S/T\right)=\frac{P\left(S\cap T\right)}{P\left(T\right)} \\ \Rightarrow P\left(S\cap T\right)=P\left(S/T\right)\times P\left(T\right)=0.4\times 0.3=0.12 \\ P\left(T/S\right)=\frac{P\left(S\cap T\right)}{P\left(S\right)}=\frac{0.12}{0.2}=0.6 \end{gathered}$$