Question

What is the probability your friend will pass the exam. There are 20 true/false items on the exam. Each question has the same weight and 60% or more correct answers results in a passing grade.

Answer 1

Consider $X$  as the number of questions answered correctly among the 20 questions: $X\sim B(n,p)$

$$P(X=x)=\binom{n}{x}p^x(1-p)^{n-x}$$

Given that $p=0.5, n=20$

$0.6\cdot20=12$

$$P(X=12)=\binom{20}{12}0.5^12(1-0.5)^{20-12}=0.120134$$

$$P(X=13)=\binom{20}{13}0.5^{13}(1-0.5)^{20-13}=0.073929$$

$$P(X=14)=\binom{20}{14}0.5^{14}(1-0.5)^{20-14}=0.036964$$

$$P(X=15)=\binom{20}{15}0.5^{15}(1-0.5)^{20-15}=0.014786$$

$$P(X=16)=\binom{20}{16}0.5^{16}(1-0.5)^{20-16}=0.004621$$

$$P(X=17)=\binom{20}{17}0.5^{17}(1-0.5)^{20-17}=0.001087$$

$$P(X=18)=\binom{20}{18}0.5^{18}(1-0.5)^{20-18}=0.000181$$

$$P(X=19)=\binom{20}{19}0.5^{19}(1-0.5)^{20-19}=0.000019$$

$$P(X=20)=\binom{20}{20}0.5^ {20}(1-0.5)^{20-20}=0.000001$$

$P(X\geq12)=P(X=12)+P(X=13)+P(X=14)+$$+P(X=15)+P(X=16)+P(X=17)+P(X=18)+$

$+P(X=19)+P(X=20)\approx$ $0.251722$

The probability my friend will pass the exam is $0.2517$ $(\approx25\%)$