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?

Answer 1

Dear Student,

$$\begin{gathered} Therearetwopossiblecasestomeetourobjectice. \\ \mathit{C}\mathit{A}\mathit{S}\mathit{E}\mathbf{\hspace{0.17em}}\mathbf{1}\mathbf{:}Getoneofthe12non-acespadesandoneoftheremaining3non-spadeaces. \\ Probability=\frac{{}^{12}C_1\times {}^{3}C_1}{{}^{52}C_2} \\ \mathit{C}\mathit{A}\mathit{S}\mathit{E}\mathbf{\hspace{0.17em}}\mathbf{2}\mathbf{:}Gettheaceofspadesandanyoneoftheother51cards. \\ Probability=\frac{{}^{1}C_1\times {}^{51}C_1}{{}^{52}C_2} \\ Requiredprobability=\frac{{}^{12}C_1\times {}^{3}C_1}{{}^{52}C_2}+\frac{{}^{1}C_1\times {}^{51}C_1}{{}^{52}C_2} \\ Requiredprobability=\frac{36}{1326}+\frac{51}{1326} \\ Requiredprobability=\frac{87}{1326}=\frac{29}{442} \end{gathered}$$

Regards