[Math] Probability in multiple choice exams

probability

  • A multiple choice exam has 175 questions.
  • Each question has 4 possible answers.
  • Only 1 answer out of the 4 possible answers is correct.

  • The pass rate for the exam is 70% (123 questions must be answered correctly).

  • We know 100 questions were answered correctly.

  • What is the probability of passing the exam, if one were to guess on the remaining 75 questions?

Best Answer

The number $X$ of correct answers in the remaining $75$ that were answered randomly, is binomially distributed with parameters $n=75$ and $p=0.25$, in symbols $$X \sim B(n=75, p=0.25)$$ The probability to pass the exam is equal to $$P(X\ge23)=\sum_{x=23}^{75}P(X=x)$$ Since it is difficult to calculate the above sum if you are not allowed to use a calculator or a program like excel, you can approximate $X$ with a normal random variable $Y$ such that $$Y \sim N(μ=np, σ^2=np(1-p)) \implies Y \sim N(μ=18.75, σ^2=14.065)$$ Now $$P(X\ge 23)=1-P(Y\le 22.5)$$ where we applied also the continuity correction.

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