+1859 A standard deck of cards contains $52$ cards. These $52$ cards are arranged in a row, at random. Find the expected number of pairs of adjacent cards that are both hearts.
+121 To find the expected number of pairs of adjacent cards that are both hearts, we can consider each position in the row of cards and calculate the probability of having two adjacent hearts at that position.
In a standard deck of 52 cards, there are 13 hearts. Since each card is equally likely to be in any position, the probability of having a heart at a specific position is \( \frac{13}{52} = \frac{1}{4} \).
Now, let's consider each position in the row. For the first position, we cannot have a pair of adjacent hearts, so the probability is 0. For the second position, the probability of having a pair of adjacent hearts is \( \frac{1}{4} \). For the third position, the probability is also \( \frac{1}{4} \). Similarly, for all positions from 1 to 51 (except the last one), the probability of having a pair of adjacent hearts is \( \frac{1}{4} \).
For the last position, we cannot have a pair of adjacent hearts again, so the probability is 0.
Since there are 51 possible positions where we can have adjacent hearts, and the probability is \( \frac{1}{4} \) for each position, the expected number of pairs of adjacent hearts is:
\[ \text{Expected number} = 51 \times \frac{1}{4} = \frac{51}{4} = 12.75. \]
Therefore, the expected number of pairs of adjacent cards that are both hearts is \( \boxed{12.75} \).
+170 For any pair of adjacent cards, the probability that they are both hearts is \(\frac{13}{52} \cdot \frac{12}{51}=\frac1{17}\). Now use linearity of expectation.
