Equal tail counts in two sets of four tosses

Question

Two people each toss a fair coin four times. Find the probability that both of them get the same number of tails.

Short answer

$35/128$ . The chance they match is $\sum_k \binom4k^2/2^8$ . By Vandermonde's identity the sum is $\binom84=70$ , giving $70/256=35/128$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Conditional Probability, Total Probability, and Bayes' Rule

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