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 11/50, then n is equal to ________ .

Solution

Maths sir · 5 steps

Split on what the lost card was, and let the n spades drawn tell you which case is more likely.

1. Two cases. Let S be the event that the lost card is a spade, with P(S) = 1/4, and S' the event that it is not, with P(S') = 3/4. Let D be the event that all n cards drawn are spades.
2. Likelihoods. If a spade is lost, 12 spades remain, so P(D | S) = C(12, n)/C(51, n). If a non-spade is lost, 13 spades remain, so P(D | S') = C(13, n)/C(51, n).

   P(S | D) = (1/4C(12, n))/(1/4C(12, n) + 3/4C(13, n)) = 1/(1 + (3 × 13)/(13-n)) = (13-n)/(52-n)

3. Simplify the ratio. The C(51, n) cancels. The step above uses C(13, n)/C(12, n) = 13/(13-n).
4. Solve for n. Set (13-n)/(52-n) = 11/50. Then 650 - 50n = 572 - 11n, so 39n = 78.
5. So. n = 2. Check: 11/50 matches the given probability.

The idea to keep. In lost-card problems, the hidden card is the cause and the cards you draw are the evidence. Write P(evidence | each cause), and the common C(51, n) always cancels.

Answer from the NTA final answer key. Solution by PrepWiser.