Method 1:
first flip:no one can win
second flip:no one can win
third flip:first player win with probability 1/4 (HHH or TTT)
fouth flip:second player win with probablility (3/4)*(1/4)
fifth flip:first play win with probablity (3/4)^2 * (1/4)
sixth flip:second play win with probablity (3/4)^3 * (1/4)
(Notice 1/4+(3/4)*(1/4)+(3/4)^2 * (1/4)+...= (1/4)*(1/(1-3/4))= 1)
P(first play wins)=(1/4)+(3/4)^2 * (1/4)+...=(1/4)* (1+(3/4)^2+(3/4)^4...)=(1/4)(1/(1-9/16))=4/7
Method 2:
We know third flip:first player win with probability 1/4 (HHH or TTT)
So second player is just like first player except there is the 1/4 chance that first player wins first.
=> P(second player win)=3/4 P(first player win)
P(second player win)+ P(first player win) = 1
=>3/4 P(first player win)+P(first player win)=1
=>P(first player win)=4/7
