One Extra Coin

I toss four fair coins and you toss five. You win if you get strictly more heads than I do. What is the probability that you win?

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#More heads or more tails, exactly one

Write $H_Y$ for your head count out of five and $H_M$ for mine out of four. You get strictly more heads when $H_Y > H_M$ . You get strictly more tails when $5 - H_Y > 4 - H_M$ , which rearranges to $H_Y \le H_M$ . These two conditions, $H_Y > H_M$ and $H_Y \le H_M$ , partition every outcome, so exactly one of them holds on any toss.

#Symmetry splits it evenly

Flipping every coin from heads to tails swaps the two events and leaves all probabilities untouched, since a fair coin treats heads and tails alike. So $\PP(\text{more heads})$ equals $\PP(\text{more tails})$ , and because the two cover all outcomes between them,

\PP(\text{you win}) = \frac{1}{2}. \tag{1}

With five coins against four, exactly one of more heads or more tails happens. Flipping every coin turns one into the other and keeps the odds the same, so each is exactly one half.

#Read it off

The answer is exactly $\tfrac{1}{2}$ , independent of how many coins are in play, as long as you toss one more than I do. That single extra coin is exactly what tips every would-be tie in your favor.