The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both black? Express your answer as a common fraction.
The number of pairs of adjacent cards which are both black is equal to the number of black cards which have another black card to their right. For each black card, there is a $\dfrac{25}{51}$ chance that the card to its right is also black, giving 1 pair, and a $\dfrac{26}{51}$ chance that the card to its right is red, giving 0 pairs. There are 26 black cards, so the expected value of the number of pairs of adjacent black cards is $$26\left(\frac{25}{51}(1)+\frac{26}{51}(0)\right)=\boxed{\frac{650}{51}}$$
\frac{650}{51}
Level 5
Counting & Probability
Problem: The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards which are both black? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the expected number of pairs of adjacent cards that are both black, we first recognize that a standard 52-card deck contains 26 black cards (spades and clubs) and 26 red cards (hearts and diamonds). 

When the cards are arranged in a circle, each card has exactly two adjacent cards. Therefore, there are a total of 52 pairs of adjacent cards in the circle. 

Next, we define an indicator random variable \(X_i\) for each pair of adjacent cards \(i\) (where \(i\) ranges from 1 to 52). The variable \(X_i\) takes the value 1 if both cards in the pair are black, and 0 otherwise. The total number of pairs of adjacent black cards can be expressed as the sum of these indicator variables: 

\[ X = X_1 + X_2 + \ldots + X_{52} \]

To find the expected value \(E[X]\), we use the linearity of expectation:

\[ E[X] = E[X_1] + E[X_2] + \ldots + E[X_{52}] \]

Now, we need to calculate \(E[X_i]\) for a single pair of adjacent cards. The probability that both cards in a specific pair are black can be calculated as follows:

1. The probability that the first card is black is \(\frac{26}{52} = \frac{1}{2}\).
2. Given that the first card is black, there are now 25 black cards left out of 51 total cards. Thus, the probability that the second card is also black is \(\frac{25}{51}\).

Therefore, the probability that both cards in the pair are black is:

\[ E[X_i] = P(\text{both cards are black}) = \frac{26}{52} \cdot \frac{25}{51} = \frac{1}{2} \cdot \frac{25}{51} = \frac{25}{102} \]

Since there are 52 pairs of adjacent cards, we multiply the expected value for one pair by the total number of pairs:

\[ E[X] = 52 \cdot E[X_i] = 52 \cdot \frac{25}{102} = \frac{1300}{102} \]

To simplify \(\frac{1300}{102}\), we find the greatest common divisor of 1300 and 102, which is 2:

\[ \frac{1300 \div 2}{102 \div 2} = \frac{650}{51} \]

Thus, the expected number of pairs of adjacent cards that are both black is \(\frac{650}{51}\).<|END_OF_LATENT|> The final answer is $\frac{650}{51}$.