When a random quantity counts many local events, indicators can simplify its expectation. The events may overlap, but expectation still adds. Independence is needed for some probability products, not for linearity itself.
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Define exactly what is counted
Toss a fair coin independently eight times. There are seven adjacent pairs of positions: 1-2 through 7-8. Count how many pairs show equal results, allowing overlaps.
For any specified pair, HH and TT account for two of four equally likely outcomes, so its probability of equality is 1/2.
Add the expected contributions
Let each pair contribute 1 when equal and 0 otherwise. Each indicator has expectation 1/2. The total expected count is therefore 7/2 = 3.5.
A single realised count is an integer. An expectation of 3.5 is an average over repeated outcomes, not a claim that one sequence contains half a pair.
Do not overstate the conclusion
Adjacent-pair events share tosses, and a separate distribution calculation would need careful dependence analysis. The expectation argument does not by itself give the probability of exactly three equal pairs.
This original exercise develops probability reasoning for AMC 12 preparation, not an official problem prediction. Define the counted objects and whether overlap is allowed before introducing indicators.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
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Official information and further reading: Source 1
