Counting head pairs
A fair coin is tossed 10 times. Count the positions , from 1 to 9, where toss and toss are both heads. Overlapping pairs count separately, so three heads in a row give 2.
Find the expected value of this count.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Write the count as a sum of 9 indicators, one per pair of neighbouring tosses.
The indicators overlap and are not independent. Does that matter for the expectation?
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Split the count into one indicator per pair of neighbouring tosses. Expectation is linear, so the dependence between overlapping pairs does not matter.
Step 1. Let if tosses and are both heads, for . The count is .
Step 2. Each pair is two independent fair tosses, so .
Step 3.
Remark. and share toss 2, so they are dependent: that matters for the variance of , not for its mean. The same argument gives for the expected number of consecutive double sixes in 100 rolls of a die.