Question

Two cards are drawn from a well shuffled pack of $52$ cards one after the other without replacement. Find the probability that one of these is a queen and the other is a king of opposite colour.

Answer 1

Since cards are drawn without replacement.

Total no. of cases, i.e., $P\left(S\right)=51\times 51=2652$

Let $A$ be the event of getting a queen and king of opposite colours

$A=\{Q_h{K}_s,Q_h{K}_c,Q_d{K}_s,Q_d{K}_c,Q_s{K}_h,Q_s{K}_d,Q_c{K}_h,Q_c{K}_d,K_h{Q}_s,K_h{Q}_c,K_d{Q}_s,K_d{Q}_c,K_s{Q}_h,K_s{Q}_d,K_c{Q}_h,K_c{Q}_d\}$

Here $h,s,d,c$ represent heart, spade, diamond and club respectively.

$\therefore n\left(A\right)=16$

$\therefore P\left(A\right)=\frac{n\left(A\right)}{n\left(S\right)}=\frac{16}{2652}=\frac{4}{663}$

Hence the probability that one of these is a queen and the other is a king of opposite colour is $\frac{4}{663}$.