Breaking Even in a Coin Game
Lewis and Lilly are playing a game with a fair coin. They each start with $\$5$. The coin will be tossed up to
0$ times. On each toss, if it lands heads, Lewis pays Lilly $\
$; if tails, Lilly pays Lewis $\
$. The game ends early if either player's balance hits $\$0$. Otherwise, it continues for all
0$ tosses.
What is the probability that both players finish the game with exactly $\$5$ -- i.e., they break even?
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