JEE MAIN - Mathematics (2025 - 4th April Evening Shift - No. 22)
A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\frac{11}{50}$, then n is equal to ________ .
Answer
2
Explanation
$P\left(\frac{\text { Lost }_{\text {(spade })}}{\mathrm{n} \text { cards are spade }}\right)$
$$\begin{aligned} & =\frac{P\left(\frac{n_s}{L_s}\right) P\left(L_s\right)}{P\left(\frac{n_s}{L_s}\right) P\left(L_s\right)+P\left(\frac{n_s}{\bar{L}_s}\right) P\left(\bar{L}_s\right)} \\ & =\frac{\frac{{ }^{12} C_n}{{ }^{51} C_n} \times \frac{1}{4}}{\frac{{ }^{12} C_n}{{ }^{51} C_n} \times \frac{1}{4}+\frac{3}{4} \times \frac{{ }^{13} C_n}{{ }^{51} C_n}}=\frac{1}{1+3 \cdot \frac{{ }^{13} C_n}{{ }^{12} C_n}}=\frac{13-n}{52-n} \\ & \Rightarrow \frac{13-n}{52-n}=\frac{11}{50} \\ & \Rightarrow n=2 \end{aligned}$$
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