Question

A student is given a test with $8$ items of true-false type. If he gets $6$ or more items correct, he is declared pass. Given that he guesses the answer to each item, compute the probability that he will pass in the test.

Answer 1

P(correct) $=\frac{1}{2}$ P(incorrect) $=\frac{1}{2}$

P(pass) $=$ P(correct $=6$ ) $+$ P(correct $=7$ ) $+$ P(correct $=8$)

$=8_{C_6}{\left(\frac{1}{2}\right)}^6{\left(\frac{1}{2}\right)}^2+8_{C_7}{\left(\frac{1}{2}\right)}^7\left(\frac{1}{2}\right)+8_{C_8}{\left(\frac{1}{2}\right)}^8{\left(\frac{1}{2}\right)}^0$

$=(8_{C_6}+8_{C_7}+8_{C_8}){\left(\frac{1}{2}\right)}^8$

$=\frac{(28+8+1)}{2^8}$

$=\frac{37}{256}$

$=0.144$