A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of n , so that the probability of guessing at least n correct answers is less than 1 2 , is :

Step-by-step solution

P E < 1 2 ⇒ ∑ r = n 8 C r 8 1 2 8 - r 1 2 r < 1 2 ⇒ ∑ r = n 8 C r 8 1 2 8 < 1 2 ⇒ C n 8 + C n + 1 8 + … + C 8 8 < 128 ⇒ 256 - C 0 8 + C 1 8 + … + C n - 1 8 < 128 ⇒ C 0 8 + C 1 8 + … + C n - 1 8 > 128 ⇒ 1 + 8 + 28 + 56 + 70 + C 5 8 + … + C n - 1 8 > 128 ⇒ 163 + C 5 8 + … + C n - 1 8 > 128 ⇒ n - 1 ≥ 4 ⇒ n ≥ 5