A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit.  Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\heartsuit$ and $\diamondsuit$) are red.  The deck is randomly arranged. What is the probability that the top card is red and the second card is black?
There are 26 ways to choose the first card to be red, then 26 ways to choose the second card to be black.  There are $52 \times 51$ ways to choose any two cards.  So the probability is $\dfrac{26 \times 26}{52 \times 51} = \boxed{\dfrac{13}{51}}$.
\dfrac{13}{51}
Level 3
Counting & Probability
Problem: A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit.  Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\heartsuit$ and $\diamondsuit$) are red.  The deck is randomly arranged. What is the probability that the top card is red and the second card is black?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability that the top card is red and the second card is black, we first identify the total number of cards in the deck, which is 52. There are 26 red cards (13 hearts and 13 diamonds) and 26 black cards (13 spades and 13 clubs).

The event of interest consists of two sequential draws without replacement. 

1. Calculate the probability of drawing a red card first. There are 26 favorable outcomes (red cards) out of 52 total cards. Thus, the probability of drawing a red card first is:
   P(Red first) = 26/52 = 1/2.

2. After drawing a red card, there are now 51 cards left in the deck. The number of black cards remains 26 since we have not drawn any black cards yet.

3. Now, we calculate the probability of drawing a black card second. The probability of drawing a black card after a red card has been drawn is:
   P(Black second | Red first) = 26/51.

4. To find the overall probability of both events occurring (drawing a red card first and then a black card), we multiply the probabilities of the two independent events:
   P(Red first and Black second) = P(Red first) * P(Black second | Red first) = (1/2) * (26/51).

5. Simplifying this expression gives:
   P(Red first and Black second) = 26/102 = 13/51.

Thus, the final answer is 13/51.<|END_OF_LATENT|> The final answer is $\dfrac{13}{51}$.