Question

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that
(i) both are kings
(ii) the first is a king and the second is an ace
(iii) the first is a heart and second is red.

Answer 1

(i) Consider the given events
A = A king in the first draw
B = A king in the second draw

$$\begin{gathered} \mathrm{Now}, \\ P\left(A\right)=\frac{4}{52}=\frac{1}{13} \\ P\left(B/A\right)=\frac{3}{51}=\frac{1}{17} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(A\cap B\right) \\ =P\left(A\right)\times P\left(B/A\right) \\ =\frac{1}{13}\times \frac{1}{17} \\ =\frac{1}{221} \end{gathered}$$

(ii) Consider the given events
A = A king in the first draw
B = An ace in the second draw

$$\begin{gathered} \mathrm{Now}, \\ P\left(A\right)=\frac{4}{52}=\frac{1}{13} \\ P\left(B/A\right)=\frac{4}{51}=\frac{4}{51} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(A\cap B\right) \\ =P\left(A\right)\times P\left(B/A\right) \\ =\frac{1}{13}\times \frac{14}{51} \\ =\frac{4}{663} \end{gathered}$$

(iii) Consider the given events.
A = A heart in the first throw
B = A red card in the second throw

$$\begin{gathered} \mathrm{Now}, \\ P\left(A\right)=\frac{13}{52}=\frac{1}{4} \\ P\left(B/A\right)=\frac{25}{51} \\ \therefore \mathrm{Required}\mathrm{probability}=P\left(A\cap B\right) \\ =P\left(A\right)\times P\left(B/A\right) \\ =\frac{1}{4}\times \frac{25}{51} \\ =\frac{25}{204} \end{gathered}$$