A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

Consider the events, <br><br>E₁ = missing card is spade <br><br>E₂ = missing card is not a spade <br><br>A = Two spade cards are drawn <br><br>$P\left( {{E_1}} \right) = {1 \over 4}$ <br>$P\left( {{E_2}} \right) = {3 \over 4}$ <br><br>$P\left( {{A \over {{E_1}}}} \right) = {{{}^{12}{C_2}} \over {{}^{51}{C_2}}}$ <br><br>$P\left( {{A \over {{E_2}}}} \right) = {{{}^{13}{C_2}} \over {{}^{51}{C_2}}}$ <br><br>$$P\left( {{{{E_2}} \over A}} \right) = {{P\left( {{A \over {{E_2}}}} \right).P\left( {{E_2}} \right)} \over {P\left( {{A \over {{E_1}}}} \right).P\left( {{E_1}} \right) + P\left( {{A \over {{E_2}}}} \right).P\left( {{E_2}} \right)}}$$ <br><br>$$ = {{{{{}^{13}{C_2}} \over {{}^{51}{C_2}}}.{3 \over 4}} \over {{{{}^{12}{C_2}} \over {{}^{51}{C_2}}}.{1 \over 4} + {{{}^{13}{C_2}} \over {{}^{51}{C_2}}}.{3 \over 4}}}$$ <br><br>= ${{39} \over {50}}$