Question
Two cards are drawn at random from a pack of $$52$$ cards. The probability of getting at least a spade and an ace is :
A.
$$\frac{1}{{34}}$$
B.
$$\frac{8}{{221}}$$
C.
$$\frac{1}{{26}}$$
D.
$$\frac{2}{{51}}$$
Answer :
$$\frac{1}{{26}}$$
Solution :
$$n\left( S \right) = {}^{52}{C_2}{\text{ and}}$$
$$n\left( E \right) = $$ the number of selections of $$1$$ spade, $$1$$ ace from $$3$$ aces or selections of the ace of spade and $$1$$ other spade
$$\eqalign{
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {}^{13}{C_1} \times {}^3{C_1} + {}^{12}{C_1} \times {}^1{C_1} = 51 \cr
& \therefore \,\,\,\,P\left( E \right) = \frac{{51}}{{{}^{52}{C_2}}} = \frac{1}{{26}}. \cr} $$