Matching tail counts in two blocks of tosses

Question

A fair coin is tossed $5$ times. What is the probability that the number of tails among tosses $1$ – $3$ equals the number of tails among tosses $4$ – $5$ ?

Short answer

$5/16$ . Match the counts $j=0,1,2$ : the number of outcomes is $\sum_j\binom3j\binom2j=1+6+3=10$ out of $32$ ; by Vandermonde's identity this sum is just $\binom52$ .

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

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Lessons behind this question

Random Variables and Probability Mass Functions
Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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