Here there are 2 cases..Let the probability of head be 'p' as given in question and that of tail be 'q'..Lets find probability for each of the cases and then we sum it up :
Case 1 : The toss begins with head :
In that case it is possible that first 2 head comes and the game is ended..Else one head , one tail then 2 heads n then game is won and so on..
So P(win in 1st case) = p2 [First two tosses lead to head] + pqp2 [First head , then tail , then two heads in a row] + pqpqp2 + ...
= p2 [ 1 + pq + (pq)2 + .......... ]
= p2 / (1 - pq) [As the internal terms form an infinite G.P with initial term = 1 and common ratio = pq]
Case 2 : Toss begins with tail :
So here it is possible that first is tail then two heads and hence game ends and results in win..Else a tail , then a head then a tail then two heads in a row and hence game ends here and so on..
Hence P(win in 2nd case) = qp2 + qpqp2 + ..........
= qp2 [ 1 + qp + (qp)2 + .......]
= qp2 / (1 - pq)
Hence P(win) = P(win in case 1) + P(win in case 2)
= [p2 / (1 - pq)] + [qp2 / (1 - pq)]
= p2 (1 + q) / (1 - pq)
Hence A) should be the correct answer..