There are 26 black(B) and 26 red(R) cards in a standard deck. A run is number of blocks of consecutive cards of the same color. For example, a sequence RRRRBBBRBRB of only 11 cards has 6 runs; namely, RRRR, BBB, R, B, R, B. Find the expected number of runs in a shuffled deck of cards.

To solve the problem of finding the expected number of runs in a shuffled deck of 52 cards (26 black and 26 red), we can use the concept of expectation in probability.

### Understanding a "Run"
A "run" in this context is defined as a sequence of consecutive cards of the same color. For example, in the sequence \( RRRRBBBRBRB \), the runs are \( RRRR \), \( BBB \), \( R \), \( B \), \( R \), \( B \), so there are 6 runs.

### Key Idea:
We need to determine the expected number of runs in a deck of 52 cards. A run can start either at the first card or whenever the color changes between two consecutive cards. 

Let \( X \) be the number of runs in the deck. We can express \( X \) as:
\[
X = 1 + \sum_{i=1}^{51} I(i)
\]
where \( I(i) \) is an indicator variable that takes the value 1 if the color of the \( (i+1) \)-th card is different from the \( i \)-th card, and 0 otherwise. The initial 1 accounts for the first run starting from the first card.

### Expected Value:
The expected value of \( X \), \( E(X) \), is given by:
\[
E(X) = 1 + \sum_{i=1}^{51} E(I(i))
\]
Since \( E(I(i)) \) is the probability that the color of the \( (i+1) \)-th card is different from the \( i \)-th card, we need to calculate \( E(I(i)) \).

For any \( i \), the probability that the color of the \( (i+1) \)-th card differs from the \( i \)-th card is \( \frac{26}{51} \). This is because, given a card of one color, there are 26 cards of the opposite color out of the remaining 51 cards.

Thus,
\[
E(X) = 1 + \sum_{i=1}^{51} \frac{26}{51}
\]
This simplifies to:
\[
E(X) = 1 + 51 \times \frac{26}{51}
\]
\[
E(X) = 1 + 26
\]
\[
E(X) = 27
\]

### Conclusion:
The expected number of runs in a shuffled deck of 52 cards (with 26 black and 26 red cards) is \( \boxed{27} \).

Would you like more details or have any further questions?

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Here are 5 relative questions to consider:
1. What is the probability of getting exactly 2 runs in a shuffled deck?
2. How would the expected number of runs change if the deck had more than two colors?
3. Can you calculate the variance of the number of runs in a shuffled deck?
4. How does the expected number of runs change if there are unequal numbers of black and red cards?
5. What is the probability that the first card and the last card in the deck belong to the same color?

**Tip:** Understanding indicator variables can greatly simplify finding expectations in problems involving sequences.

Explore the expected number of runs in a shuffled deck of 52 cards with 26 black and 26 red cards. Understand the concept of runs, probability calculations, and indicator variables used to determine the expected value. Suitable for advanced high school students and above.

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