Expected consecutive pairs

What this expected consecutive pairs question tests

This is a medium-difficulty probability question that combines combinatorial sampling with expectation calculation. It appears frequently in quant interview loops to assess whether a candidate can leverage linearity of expectation to transform a seemingly complex counting problem into a tractable one.

The core skill is recognizing that you don't need to enumerate all possible 6-element subsets. Instead, you can decompose the total count into indicator random variables for each potential consecutive pair, compute the probability that each forms a consecutive pair in your sample, and sum them up. This approach sidesteps messy casework and makes the calculation direct and verifiable.

• Linearity of expectation over indicator random variables
• Hypergeometric reasoning (sampling without replacement)
• Symmetry and identical probability for structurally similar events