A student appeared in an examination consisting of 8 true-false type questions.

Question

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 $${1 \over 2}$$, is :

Accepted Answer

$$P(E) < {1 \over 2}$$

$$ \Rightarrow \sum\limits_{r = n}^8 {{}^8{C_r}} {\left( {{1 \over 2}} \right)^{8 - r}}{\left( {{1 \over 2}} \right)^r} < {1 \over 2}$$

$$ \Rightarrow \sum\limits_{r = n}^8 {{}^8{C_r}} {\left( {{1 \over 2}} \right)^8} < {1 \over 2}$$

$$ \Rightarrow {}^8{C_n} + {}^8{C_{n + 1}} + .... + {}^8{C_8} < 128$$

$$ \Rightarrow 256 - \left( {{}^8{C_0} + {}^8{C_1} + ... + {}^8{C_{n - 1}}} \right) < 128$$

$$ \Rightarrow {}^8{C_0} + {}^8{C_1} + .... + {}^8{C_{n - 1}} > 128$$

$$ \Rightarrow n - 1 \ge 4$$

$$ \Rightarrow n \ge 5$$

Solutions · 1