TITLE: Probability Proof: Same Probability in Coin Toss
QUESTION [2 upvotes]: Given a fair coin, suppose that the first friend flips the coin $k$ times and the second friend flips the coin $n-k$ times. How can we prove that the probability that both friends flip the same number of heads is the same as the probability that there are total of $k$ heads?

REPLY [1 votes]: For $i=1,2$ let $H_i$ denote the number of heads and let $T_i$ denote the number of tails thrown by friend $i$. 
Then $\langle H_1,H_2\rangle$ and $\langle T_1,H_2\rangle$ have equal distributions so that:
$$P(H_1+H_2=k)=P(T_1+H_2=k)=P(k-H_1+H_2=k)=P(H_1=H_2)$$
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