GATE CS 1996 Q2.7 — Probability: Random Variables and Distributions

A standard deck of cards contains $52$ cards, of which $4$ are aces.

We want to find the probability that both the top card and the bottom card of a randomly shuffled deck are aces. This can be calculated step-by-step using conditional probability:

1. Probability that the top card is an ace:
   There are $4$ aces out of $52$ total cards.
   $$P(\text{Top is Ace}) = \frac{4}{52}$$
2. Probability that the bottom card is an ace, given the top card is an ace:
   After placing an ace at the top, there are $51$ cards remaining in the deck, of which $3$ are aces.
   $$P(\text{Bottom is Ace} \mid \text{Top is Ace}) = \frac{3}{51}$$

Using the multiplication rule of probability, the joint probability that both the top and bottom cards are aces is:

Hence, the correct option is C.