Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that they are both aces if the first card is replaced.
The probability of drawing an ace from a well shuffled pack of 52 cards is $\dfrac{4}{52}=\dfrac{1}{13}$. Since the first card is replaced, the probability of drawing a second ace is again $\dfrac{1}{13}$. Therefore, the probability that both cards are aces is $\left(\dfrac{1}{13}\right)^2=\boxed{\dfrac{1}{169}}$.
To find the probability that two cards drawn from a well-shuffled pack of 52 cards are both aces, given that the first card is replaced, we need to consider the following:
Step 1: Determine the probability of drawing an ace on the first draw. Since there are 4 aces in a deck of 52 cards, the probability of drawing an ace on the first draw is 4/52 (or 1/13).
Step 2: Since the first card is replaced, the deck is returned to its initial state, and the probability of drawing an ace on the second draw is also 4/52 (since there are still 4 aces remaining in the deck).
Step 3: To find the probability of both events occurring, we multiply the individual probabilities together. Hence, the probability of drawing two aces in a row (with replacement on the first card) is (1/13) * (1/13).
Simplifying this expression, we find that the probability of drawing two aces in a row, with replacement, is 1/169.
Therefore, the probability that two cards drawn are both aces, given that the first card is replaced, is 1/169.