Question

homework 32: problem 6 (1 point) a quiz consists of 20 multiple - choice questions, each with 5 possible answers. for someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 60 %. p(pass)=□ you have attempted this problem 6 times. your overall recorded score is 0 %. you have unlimited attempts remaining. preview my answers submit answers show past answers

Explanation:

Step1: Identify probability of correct answer per question

The probability of getting a single - multiple - choice question (with 5 possible answers) correct by random guessing is $p=\frac{1}{5}=0.2$, and the probability of getting it wrong is $q = 1 - p=1 - 0.2 = 0.8$.

Step2: Determine the number of questions needed to pass

To pass with a 60% grade on a 20 - question quiz, the number of correct answers $x$ must satisfy $\frac{x}{20}\geq0.6$, so $x\geq12$.

Step3: Use the binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times q^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, $q$ is the probability of failure on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. We want $P(X\geq12)=\sum_{k = 12}^{20}C(20,k)\times(0.2)^{k}\times(0.8)^{20 - k}$.
$C(20,k)=\frac{20!}{k!(20 - k)!}$.
$P(X = 12)=\frac{20!}{12!(20 - 12)!}\times(0.2)^{12}\times(0.8)^{8}$
$P(X = 13)=\frac{20!}{13!(20 - 13)!}\times(0.2)^{13}\times(0.8)^{7}$
$\cdots$
$P(X = 20)=\frac{20!}{20!(20 - 20)!}\times(0.2)^{20}\times(0.8)^{0}$
Calculating each term:
$C(20,12)=\frac{20!}{12!8!}=\frac{20\times19\times18\times17\times16\times15\times14\times13}{8\times7\times6\times5\times4\times3\times2\times1}=125970$
$P(X = 12)=125970\times(0.2)^{12}\times(0.8)^{8}\approx125970\times5.9604644775390625\times10^{-9}\times0.16777216\approx0.00012$

After calculating and summing up $P(X = k)$ for $k$ from 12 to 20:
$P(X\geq12)\approx0$

Answer:

$0$