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 being a spade is $\tfrac{11}{50}$, then n is equal to:

Step 1: Define Events and Probabilities

Let $L_S$ be the event that the lost card is a spade, and $L_{NS}$ be the event that the lost card is not a spade. A standard deck has 13 spades and 39 non-spades. We calculate the initial probabilities for these events.

Step 2: Calculate Conditional Probabilities of Drawing Spades

Let $E$ be the event that $n$ cards drawn from the remaining 51 cards are all spades. If the lost card was a spade ($L_S$), there are 12 spades left in the 51 cards. If the lost card was not a spade ($L_{NS}$), there are 13 spades left in the 51 cards. We use combinations to find the probability of drawing $n$ spades in each case.

Step 3: Apply Bayes' Theorem

We are given the probability of the lost card being a spade, given that $n$ cards drawn are spades, which is $P(L_S|E) = \frac{11}{50}$. We use Bayes' theorem to relate this to the probabilities we calculated.

Step 4: Substitute and Simplify

Substitute the probabilities into Bayes' theorem. The term $\frac{1}{4\binom{51}{n}}$ cancels out from the numerator and denominator, simplifying the expression.

Step 5: Expand Combinations and Solve for n

We expand the combination terms using the formula $\binom{k}{r} = \frac{k!}{r!(k-r)!}$. After simplification, we get an equation in terms of $n$. We then solve this algebraic equation to find the value of $n$.

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