Let the probability of getting head for a biased coin be 1 4 . It is tossed repeatedly until a head appears. Let N be the number of tosses required. If the probability that the equation 64 x 2 + 5 N x + 1 = 0 has no real root is p q , where p and q are co-prime, then q - p is equal to..........

Step-by-step solution

Given a quadratic equation 64 x 2 + 5 n x + 1 = 0 has no real roots, so D < 0 ⇒ 25 N 2 - 4 × 64 < 0 ⇒ N < 16 5 ⇒ N = 1 , 2 , 3 And, For a biased coin, the probability of getting head is P = 1 4 , and probability of getting tail will be   Q = 3 4 So, required probability is 1 4 + 3 4 · 1 4 + 3 4 2 · 1 4 + 3 4 3 · 1 4 = p q ⇒ 1 4 1 - 3 4 3 1 - 3 4 = p q ⇒ 1 - 3 4 3 = p q ⇒ 1 - 27 64 = p q ⇒ p q = 37 64 q - p = 27