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 :
Solution
$P(E) < {1 \over 2}$<br><br>$$ \Rightarrow \sum\limits_{r = n}^8 {{}^8{C_r}} {\left( {{1 \over 2}} \right)^{8 - r}}{\left( {{1 \over 2}} \right)^r} < {1 \over 2}$$<br><br>$$ \Rightarrow \sum\limits_{r = n}^8 {{}^8{C_r}} {\left( {{1 \over 2}} \right)^8} < {1 \over 2}$$<br><br>$\Rightarrow {}^8{C_n} + {}^8{C_{n + 1}} + .... + {}^8{C_8} < 128$<br><br>$$ \Rightarrow 256 - \left( {{}^8{C_0} + {}^8{C_1} + ... + {}^8{C_{n - 1}}} \right) < 128$$<br><br>$\Rightarrow {}^8{C_0} + {}^8{C_1} + .... + {}^8{C_{n - 1}} > 128$<br><br>$\Rightarrow n - 1 \ge 4$<br><br>$\Rightarrow n \ge 5$
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
This question is part of PrepWiser's free JEE Main question bank. 143 more solved questions on Probability are available — start with the harder ones if your accuracy is >70%.