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 , then n is equal to ___________. [2025]

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Solution

Q.
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 ___________. [2025]

Ans.
(2)

Let us define the events:

A : Lost card is a spade

B : n cards drawn from 51 cards are spades.

$∴ P (A) = \frac{13}{52} = \frac{1}{4}, P (A^{c}) = 1 - \frac{1}{4} = \frac{3}{4}$

$P (B / A) = \frac{{}^{12}C_{n}}{{}^{51}C_{n}}, P (B / A^{c}) = \frac{{}^{13}C_{n}}{{}^{51}C_{n}}$

$P (B) = P (A) P (B / A) + P (A^{c}) P (B / A^{c})$

$= \frac{1}{4} \frac{{}^{12}C_{n}}{{}^{51}C_{n}} + \frac{3}{4} \frac{{}^{13}C_{n}}{{}^{51}C_{n}}$

$∴ P (\frac{A}{B}) = \frac{P (B / A) P (A)}{P (B)} \Rightarrow \frac{11}{50} = \frac{\frac{{}^{12}C_{n}}{{}^{51}C_{n}} \times \frac{1}{4}}{P (B)}$

$\Rightarrow \frac{13 - n}{52 - n} = \frac{11}{50} \Rightarrow n = 2$ .

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