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

A single MCSA question with $a$ alternative answers, contains $1$ correct answer and $\left(a-1\right)$ 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:
$P\left(q=n\right)={P\left(s=1\right)}^{n}={\left(\frac{1}{a}\right)}^{n}$

• 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:
$P\left(q>x\cdot n\right)=\sum _{k=⌈x\cdot n⌉}^{n}P\left(q=k\right)=\sum _{k=⌈x\cdot n⌉}^{n}\left(\begin{array}{c}n\\ k\end{array}\right)\cdot \text{}\frac{{\left(a-1\right)}^{n-k}}{{a}^{n}}$
$\left(0\le x\le 1\right)$

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:
$P\left(q>n\cdot a\right)=\sum _{k=⌈x\cdot n\cdot a⌉}^{n\cdot a}P\left(q=k\right)=\sum _{k=⌈x\cdot n\cdot a⌉}^{n\cdot a}\left(\begin{array}{c}n\cdot a\\ k\end{array}\right)\cdot {\left(\frac{1}{2}\right)}^{n\cdot a}$
$\left(0\le x\le 1\right)$

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:
$P\left(q>x\cdot n\right)=\sum _{k=⌈x\cdot n⌉}^{n}P\left(q=k\right)=\left(\begin{array}{c}n\\ k\end{array}\right)\cdot {\left(\frac{1}{2}\right)}^{a\cdot k}\cdot {\left(1-{\left(\frac{1}{2}\right)}^{a}\right)}^{n-k}$
$\left(0\le x\le 1\right)$

Examples

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

all wrong $>\frac{1}{2}$ correct $>\frac{2}{3}$ correct $>\frac{3}{4}$ correct $>\frac{4}{5}$ correct all correct
MCSA $\frac{1}{28.42}$ $\frac{1}{235.86}$ $\frac{1}{8,831.92}$ $\frac{1}{988,871.98}$ $\frac{1}{988,871.98}$ $\frac{1}{30,517,578,125}$
MCMA partial $\frac{1}{37,778,931,862,957,161,709,568}$ $\frac{1}{2}$ $\frac{1}{382.54}$ $\frac{1}{276,084.59}$ $\frac{1}{12,596,052.18}$ $\frac{1}{37,778,931,862,957,161,709,568}$
MCMA non-partial $\frac{1}{1.61}$ $\frac{1}{208,064,843.35}$ $\frac{1}{433,008,250,196.35}$ $\frac{1}{2,766,415,372,899,402.57}$ $\frac{1}{2,766,415,372,899,402.57}$ $\frac{1}{37,778,931,862,957,161,709,568}$