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EEJ MAIN Mathematics QUESTION #6977
Question 1

A test paper has 5 questions, each with three answer choices of which only one is correct. What is the probability that a student scoring purely by guessing gets 4 or more questions right?

  • $\dfrac{17}{3^5}$
  • $\dfrac{13}{3^5}$
  • $\dfrac{11}{3^5}$✔️
  • $\dfrac{10}{3^5}$
Correct Answer Explanation

This is a binomial distribution with $n=5$, $p=\dfrac{1}{3}$ (correct), $q=\dfrac{2}{3}$ (wrong).

$P(X \geq 4) = P(X=4) + P(X=5)$

$P(X=4) = \binom{5}{4}\left(\dfrac{1}{3}\right)^4\left(\dfrac{2}{3}\right)^1 = 5 \cdot \dfrac{1}{81} \cdot \dfrac{2}{3} = \dfrac{10}{243}$

$P(X=5) = \binom{5}{5}\left(\dfrac{1}{3}\right)^5 = \dfrac{1}{243}$

$P(X \geq 4) = \dfrac{10}{243} + \dfrac{1}{243} = \dfrac{11}{243} = \dfrac{11}{3^5}$