Would anyone have a suggestion for this math/probability problem, please?

Peter and Paul bet one dollar each on each game. Each is willing
to allow the other unlimited credit. Use a calculator to make a
table showing, to four decimal places, for each of p = 1/10, ⅓,
.49, .499, .501, .51, ⅔, 9/10 the probabilities that Peter is ever
ahead by $10, by $100, and by $1000. (p is the probability of Peter winning the game)
——————-
My idea is this:
If the amounts are won by winning streaks, then the solving would be raising each p to the 10, 100, and 1000.
But one can win 10 games haltingly: e.g., win seven in a row and lose two in a row, then win five in a row.
And this is in no way possible to tabulate.

What do flutherites say?
Thank you in advance for any input.