Answer to Question #287095 in Statistics and Probability for fana

Question

Find the appropriate distribution for this question. There 20 multiple choice questions in a Statistic class quiz. Each question has 4 possible answers then only 1 of them has correct answer. Find the probability of having 3 or less correct answer if a student has to answer every question in random. Give an explanation on the result.

Answer 1

The probability of guessing the correct answer out of 4 answers is ¼ or 0.25. That makes the probability of guessing the wrong answer ¾ or 0.75. Assuming the 20 questions are independent of each other, this problem is represented by a binomial probability distribution where n = 20, p = ¼, q = ¾.

for binomial probability distribution:

$P(x=k)=C^k_n p^k q^{n-k}$

Hence, the probability of getting 3 or less correct answer:

$P(x\le 3)=P(0)+P(1)+P(2)+P(3)$

where

$P(0)=0.75^{20}=0.0032$

$P(1)=20\cdot0.25\cdot 0.75^{19}=0.0211$

$P(2)=190\cdot0.25^2\cdot 0.75^{18}=0.0167$

$P(3)=\frac{20!}{17!3!}\cdot 0.25^3 \cdot 0.75^{17}=0.1339$

$P(x\le 3)=0.0032+0.0211+0.0167+0.1339=0.1749$