Expected number of same-colour blocks in 25 cells

Question

Twenty-five cells are arranged in a row, and each is painted black or white independently with probability $\tfrac12$ each. A block is a maximal run of neighbouring cells sharing one colour; for instance BBWBWWWBBW consists of $6$ blocks. What is the expected number of blocks?

Short answer

$13$ . The number of blocks is $1$ plus the number of adjacent pairs whose colours differ. Each of the $24$ adjacent pairs differs with probability $\tfrac12$ , so by linearity the expectation is $1+24\cdot\tfrac12=13$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

See the hint and the full worked solution in the question bank (free account).

Open the worked solution

Lessons behind this question

Expectation, Mean, and Variance
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

Start with a free lesson: Expectation, Variance and Covariance is part of a free topic. Create a free account, no card needed, to read it in full with worked solutions.

Read it free