Find the Expected Number of Consecutive Chosen Pairs
Company: Sig
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Online Assessment
# Find the Expected Number of Consecutive Chosen Pairs
Choose 6 distinct numbers uniformly from the integers `1` through `35`. A consecutive pair is an adjacent integer pair for which both values were selected. For example, the selection `{1, 10, 11, 12, 20, 35}` contains the two pairs `(10, 11)` and `(11, 12)`.
What is the expected number of consecutive pairs? Express the answer as a fraction in simplest form.
### What a Strong Answer Covers
- One indicator variable for each of the 34 possible adjacent pairs.
- The probability that both members of a fixed pair are in a uniform six-element subset.
- Linearity of expectation without an independence assumption between pairs.
- Correct simplification of the final sum.
### Follow-up Questions
- What is the expected number when choosing `k` values from `1` through `n`?
- Why are indicators for overlapping pairs dependent even though linearity still applies?
Overview: Find the expected number of adjacent integer pairs in a uniformly chosen six-element subset of 1 through 35. The solution uses 34 indicators, computes each pair's inclusion probability, applies linearity despite dependence, and simplifies the expectation to 6/7.
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