Question

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 $\frac{11}{50}, \text{ then } n \text{ is equal to}$ _______

Hint:

When calculating probabilities in combinatorics, use the combination formula and adjust the number of favorable and total outcomes accordingly.

Answer 1

Correct Answer: 2

Let \( n \) cards be drawn and found to be spades. The number of spades remaining is \( 13 - x \), where \( x \) is the number of spades drawn.
Therefore, the remaining total number of cards is \( 52 - x \). We are given the probability of the lost card being a spade as \( \frac{11}{50} \). This probability can be written as: \[ P(\text{lost card is spade}) = \frac{\binom{13 - x}{1}}{\binom{52 - x}{1}} = \frac{11}{50} \] Solving this equation for \( x \), we find that \( x = 2 \), so the number of cards drawn is \( n = 2 \).
Thus, the correct answer is \( 2 \).

Answer 2

Step 1: Given that \( n \) cards are drawn and all found to be spades, the remaining number of spades is \( 13 - x \), where \( x \) is the number of spades drawn. The remaining total number of cards is \( 52 - x \). 

Step 2: Now, given that the probability \( P(\text{lost card is spade}) = \frac{11}{50} \), we can set up the following equation:

\[ \frac{\binom{13 - n}{1}}{\binom{52 - n}{1}} = \frac{11}{50} \]

Step 3: This simplifies to:

\[ 50(13 - n) = 11(52 - n) \]

Step 4: Solving the equation:

\[ 39n = 78 \] \[ n = 2 \]

Conclusion: The value of \( n \) is 2.