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 :

Solution

Maths sir · 3 steps

Out of 256 equally likely answer sheets, find the first n where at least n correct covers fewer than 128.

1. Set up. Each guess is right with probability 1/2, so the number of correct answers X follows B(8, 1/2). Each of the 2⁸ = 256 answer patterns is equally likely, so P(X = k) = C(8, k)/256.

   P(X ≥ n) = 1/256Σₖ₌ₙ⁸C(8, k) < 1/2 ⇔ Σₖ₌ₙ⁸C(8, k) < 128

2. Coefficients. For k = 3, 4, 5, 6, 7, 8 the values of C(8, k) are 56, 70, 56, 28, 8, 1. Add them from the top end down.

   n	Σ_(k ≥ n)C(8, k)	P(X ≥ n)	Less than 1/2?
   3	56 + 70 + 56 + 28 + 8 + 1 = 219	219/256	No
   4	70 + 56 + 28 + 8 + 1 = 163	163/256	No
   5	56 + 28 + 8 + 1 = 93	93/256	Yes
   6	28 + 8 + 1 = 37	37/256	Yes

3. So. The probability first drops below 1/2 at n = 5, where it is 93/256. The smallest such n is 5, option (a).

The idea to keep. With p = 1/2, every outcome is a count of C(n, k) out of 2ⁿ. Turn the probability condition into a condition on that count, then add coefficients from the end you need.

Answer from the NTA final answer key. Solution by PrepWiser.