Does that mean you can win the game 4-3?

I once solved a math monthly that asked to maximize teams winning a 7 games series.

This is the same idea. First to four wins.

Then I did calculus to show a max wrt p.

You just need to go through the possibilities.

since the loss can come in any game EXCEPT the fifth game (geometric)

since the loss can come in any game EXCEPT the sixth game (neg binomial)

we need that here

since the loss can come in any game EXCEPT the sixth game (neg binomial)

we need that as a coefficient in this one.

Now, add these, since they are mutually exclusive.