Question

A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?
(i) A = The card drawn is a king or queen, B = the card drawn is a queen or jack
(ii) A = the card drawn is black, B = the card drawn is a king
(iii) B = the card drawn is a spade, B = the card drawn in an ace

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)P\left(\text{king or queen}\right)=P\left(A\right)=\frac{8}{52}=\frac{2}{13} \\ P\left(\text{queen or jack}\right)=P\left(B\right)=\frac{8}{52}=\frac{2}{13} \\ P\left(A\cap B\right)=P\left(\text{queen}\right)=\frac{4}{52}=\frac{1}{13} \\ P\left(A\cap B\right)\ne P\left(A\right)P\left(B\right) \\ \text{Thus, A and B are not independent events.} \\ \left(\mathrm{ii}\right)P\left(\text{black}\right)=P\left(A\right)=\frac{26}{52}=\frac{1}{2} \\ P\left(\text{king}\right)=P\left(B\right)=\frac{4}{52}=\frac{1}{13} \\ P\left(A\cap B\right)=P\left(\text{black king}\right)=\frac{2}{52}=\frac{1}{26} \\ P\left(A\cap B\right)=P\left(A\right)P\left(B\right) \\ \text{Thus, A and B are independent events.} \\ \left(\mathrm{iii}\right)P\left(\text{spade}\right)=P\left(A\right)=\frac{13}{52}=\frac{1}{4} \\ P\left(\text{ace}\right)=P\left(B\right)=\frac{4}{52}=\frac{1}{13} \\ P\left(A\cap B\right)=P\left(\text{ace of spade}\right)=\frac{1}{52} \\ P\left(A\cap B\right)=P\left(A\right)P\left(B\right) \\ \text{Thus, A and B are independent events.} \end{gathered}$$