Answers Probability

This document illustrates the probabilities to randomly guess the correct answers in a multi choice questions test.

MCSA (Multiple Choice Single Answer)

A single MCSA question with $a$ alternative answers, contains $1$ correct answer and $(a-1)$ wrong answers

• the probability that a randomly selected $s$ answer is the correct one is:
  $P\left(s=1\right)=\frac{1}{a}$

• the probability that a randomly selected $s$ answer is the wrong one is:
  $P\left(s=0\right)=1-P\left(s=1\right)=1-\frac{1}{a}=\frac{a-1}{a}$

In a test composed of $n$ independent MCSA questions containing $a$ alternative answers each,

• the probability to randomly select exactly $q=k$ correct answers is:
  $P\left(q=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot {P\left(s=1\right)}^k\cdot {P\left(s=0\right)}^{n-k}=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot {\left(\frac{1}{a}\right)}^k\cdot {\left(\frac{a-1}{a}\right)}^{n-k}=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot \frac{{\left(a-1\right)}^{n-k}}{a^n}$
  $\left(0\le k\le n\right)$

• the probability to randomly select all the correct answers $\left(k=n\right)$ is:
  $P\left(q=n\right)={P\left(s=1\right)}^n={\left(\frac{1}{a}\right)}^n$

  </mrow> <!--annotation encoding="TeX"> P(q=n)=P(s=1)^n=(\frac{1}{a})^n </annotation--> 

• the probability to randomly select all the wrong answers $\left(k=0\right)$ is:
  $P\left(q=0\right)={P\left(s=0\right)}^n={\left(\frac{a-1}{a}\right)}^n$

• the probability to randomly select more than $x\cdot n$ correct answers is:
  $P\left(q>x\cdot n\right)=\sum _{k=\lceil x\cdot n\rceil }^{n}P\left(q=k\right)=\sum _{k=\lceil x\cdot n\rceil }^n\left(\begin{array}{c}n\\ k\end{array}\right)\cdot \frac{{\left(a-1\right)}^{n-k}}{a^n}$
  $\left(0\le x\le 1\right)$

MCMA (Multiple Choice Multiple Answers) with partial score option

On a single MCMA question, when the "partial score for MCMA" option is set, each individual alternative answer can be considered an independent true/false question.

A MCMA test with $n$ questions and $a$ alternative answers for each question is equivalent to a test composed of $\left(n\cdot a\right)$ MCSA questions with $2$ alternative answers each.

• the probability to randomly select exactly $q=k$ correct answers is:
  $P\left(q=k\right)=\left(\begin{array}{c}n\cdot a\\ k\end{array}\right)\cdot {\left(\frac{1}{2}\right)}^{n\cdot a}$
  $\left(0\le k\le n\cdot a\right)$

• the probability to randomly select all the correct answers $\left(k=n\cdot a\right)$ is equivalento to the probability to randomly select all the wrong answers $\left(k=0\right)$:
  $P\left(q=n\cdot a\right)=P\left(q=0\right)={\left(\frac{1}{2}\right)}^{n\cdot a}$

• the probability to randomly select more than $x\cdot n\cdot a$ correct answers is:
  $P\left(q>n\cdot a\right)=\sum _{k=\lceil x\cdot n\cdot a\rceil }^{n\cdot a}P\left(q=k\right)=\sum _{k=\lceil x\cdot n\cdot a\rceil }^{n\cdot a}\left(\begin{array}{c}n\cdot a\\ k\end{array}\right)\cdot {\left(\frac{1}{2}\right)}^{n\cdot a}$
  $\left(0\le x\le 1\right)$

MCMA (Multiple Choice Multiple Answers) without partial score option

On a single MCMA question with $a$ alternative answers, when the "partial score for MCMA" option is disabled, a question is considered correctly answered only when all the a alternative answers are set to the correct value.

• the probability that a randomly selected $s$ answer is the correct one is:
  $P\left(s=1\right)={\left(\frac{1}{2}\right)}^a$

• the probability that a randomly selected $s$ answer is the wrong one is:
  $P\left(s=0\right)=1-P\left(s=1\right)=1-{\left(\frac{1}{2}\right)}^a$

In a test composed of $n$ independent MCMA questions containing $a$ alternative answers each,

• the probability to randomly select exactly $q=k$ correct answers is:
  $P\left(q=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot {P\left(s=1\right)}^k\cdot {P\left(s=0\right)}^{n-k}=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot {\left(\frac{1}{2}\right)}^{a\cdot k}\cdot {\left(1-{\left(\frac{1}{2}\right)}^a\right)}^{n-k}$
  $\left(0\le k\le n\right)$

• the probability to randomly select all the correct answers $\left(k=n\right)$ is:
  $P\left(q=n\right)={P\left(s=1\right)}^n={\left(\frac{1}{2}\right)}^{a\cdot n}$

• the probability to randomly select all the wrong answers $\left(k=0\right)$ is:
  $P\left(q=0\right)={P\left(s=0\right)}^n={\left(1-{\left(\frac{1}{2}\right)}^a\right)}^n$

• the probability to randomly select more than $x\cdot n$ correct answers is:
  $P\left(q>x\cdot n\right)=\sum _{k=\lceil x\cdot n\rceil }^{n}P\left(q=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot {\left(\frac{1}{2}\right)}^{a\cdot k}\cdot {\left(1-{\left(\frac{1}{2}\right)}^a\right)}^{n-k}$
  $\left(0\le x\le 1\right)$

Examples

Probabilities for a test composed of $n=15$ questions containing $a=5$ alternative answers each.