Prob of Head = 0.72
Prob of Tail = 0.28
Lets see how Bob can win,
In first step : "TH"
In second step : "HH TH" or "TT TH"
In third step : " HH HH TH " or " HH TT TH" or " TT TT TH " or "TT HH TH"
Now the pattern is easy to see, we even can write a regular expression for this : (HH + TT)* TH :)
P(Bob wins)
= TH + (HH + TT)TH + (HH + TT)2TH+..................
= TH [1 + (HH + TT) + (HH + TT)2 + ................... ]
=TH/(1-((HH + TT)) [Applied Infinite GP sum = a/1-r where a is first term and r is common ration]
Now subtitute values,
P(TH) = 0.28 * 0.72
P(HH) = (0.72)2
P(TT) = (0.28)2
On solving P(Bob wins) = 0.4897