JEE MAIN - Mathematics (2021 - 16th March Morning Shift - No. 13)
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 :
$${{39} \over {50}}$$
$${{3} \over {4}}$$
$${{22} \over {425}}$$
$${{52} \over {867}}$$
Explanation
Consider the events,
E1 = missing card is spade
E2 = missing card is not a spade
A = Two spade cards are drawn
$$P\left( {{E_1}} \right) = {1 \over 4}$$
$$P\left( {{E_2}} \right) = {3 \over 4}$$
$$P\left( {{A \over {{E_1}}}} \right) = {{{}^{12}{C_2}} \over {{}^{51}{C_2}}}$$
$$P\left( {{A \over {{E_2}}}} \right) = {{{}^{13}{C_2}} \over {{}^{51}{C_2}}}$$
$$P\left( {{{{E_2}} \over A}} \right) = {{P\left( {{A \over {{E_2}}}} \right).P\left( {{E_2}} \right)} \over {P\left( {{A \over {{E_1}}}} \right).P\left( {{E_1}} \right) + P\left( {{A \over {{E_2}}}} \right).P\left( {{E_2}} \right)}}$$
$$ = {{{{{}^{13}{C_2}} \over {{}^{51}{C_2}}}.{3 \over 4}} \over {{{{}^{12}{C_2}} \over {{}^{51}{C_2}}}.{1 \over 4} + {{{}^{13}{C_2}} \over {{}^{51}{C_2}}}.{3 \over 4}}}$$
= $${{39} \over {50}}$$
E1 = missing card is spade
E2 = missing card is not a spade
A = Two spade cards are drawn
$$P\left( {{E_1}} \right) = {1 \over 4}$$
$$P\left( {{E_2}} \right) = {3 \over 4}$$
$$P\left( {{A \over {{E_1}}}} \right) = {{{}^{12}{C_2}} \over {{}^{51}{C_2}}}$$
$$P\left( {{A \over {{E_2}}}} \right) = {{{}^{13}{C_2}} \over {{}^{51}{C_2}}}$$
$$P\left( {{{{E_2}} \over A}} \right) = {{P\left( {{A \over {{E_2}}}} \right).P\left( {{E_2}} \right)} \over {P\left( {{A \over {{E_1}}}} \right).P\left( {{E_1}} \right) + P\left( {{A \over {{E_2}}}} \right).P\left( {{E_2}} \right)}}$$
$$ = {{{{{}^{13}{C_2}} \over {{}^{51}{C_2}}}.{3 \over 4}} \over {{{{}^{12}{C_2}} \over {{}^{51}{C_2}}}.{1 \over 4} + {{{}^{13}{C_2}} \over {{}^{51}{C_2}}}.{3 \over 4}}}$$
= $${{39} \over {50}}$$
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