Vipin and Shubham are playing a game with a coin, that comes up heads with a probability p . They take turns flipping the coin until one of them wins, with Vipin going first. Vipin wins if he flips a heads and Shubham wins if he flips a tails. Given that the probability of Vipin winning the game is 1 2 , then the value of p is k s i n 2 162 o . The value of k 10 is equal to

Step-by-step solution

P H = p ,    P T = 1 - p Probability of Vipin winning game is P H + P T P H P H + P T P H P T P H P H + . . . . . . . i n f i n i t e   s e r i e s = P H 1 - P H P ( T ) = p 1 - p ( 1 - p ) = p 1 - p + p 2 ⇒ p 1 - p + p 2 = 1 2   given ⇒ 2 p = 1 - p + p 2 ⇒ p 2 - 3 p + 1 = 0 ⇒ p = 3 ± 5 2   ∴ p = 3 + 5 2 > 1 ⇒ p = 3 - 5 2 = 5 - 1 4 2 × 4 = 4 s i n 2 162 o ⇒ k = 4