Question

An objective-type test paper has $5$ questions. Out of these $5$ questions, $3$ questions have four options each$(A,B,C,D)$with one option being the correct answer. The other $2$ questions have two options each, namely True and False. A candidate randomly ticks the options. Then the probability that he/she will tick the correct option in at least four questions is

Answer 1

Step 1: Calculate the probability that the student will tick exactly four questions correctly

The total number of questions with four options, $n_1=3$.

The probability that the student correctly ticks a question with four options is $p=\frac{1}{4}$.

The probability that the student incorrectly ticks a question with four options is $q=1-\frac{1}{4}=\frac{3}{4}$.

The total number of true-false questions, $n_2=2$.

The probability that the student correctly ticks a true-false question is $p=\frac{1}{2}$.

The probability that the student incorrectly ticks a true-false question is $q=1-\frac{1}{2}=\frac{1}{2}$.

Case 1: Let us assume that the student answered $3$ MCQ-type questions and $1$ True-False question correctly.

Thus, the probability of the above-mentioned case can be given by,

$$\begin{gathered} {}^{3}C_3{\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^{3-3}\times {}^{2}C_1{\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^{2-1} \\ =1\times \frac{1}{64}\times 1\times 2\times \frac{1}{2}\times \frac{1}{2} \\ =\frac{1}{128} \end{gathered}$$

Case 2: Let us assume that the student answered $2$ MCQ-type questions and $2$ True-False questions correctly.

Thus, the probability of the above-mentioned case can be given by,

$$\begin{gathered} {}^{3}C_2{\left(\frac{1}{4}\right)}^2{\left(\frac{3}{4}\right)}^{3-2}\times {}^{2}C_2{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^{2-2} \\ =3\times \frac{1}{16}\times \frac{3}{4}\times 1\times \frac{1}{4}\times 1 \\ =\frac{9}{256} \end{gathered}$$

Therefore, the probability that the student will tick exactly four questions correctly is $\frac{1}{128}+\frac{9}{256}=\frac{11}{256}$.

Step 2: Calculate the probability that the student will tick exactly five questions correctly

Let us assume that the student answered $3$ MCQ-type questions and $2$ True-False questions correctly.

Thus, the probability of the above-mentioned case can be given by,

$$\begin{gathered} {}^{3}C_3{\left(\frac{1}{4}\right)}^3{\left(\frac{3}{4}\right)}^{3-3}\times {}^{2}C_2{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^{2-2} \\ =1\times \frac{1}{64}\times 1\times 1\times \frac{1}{4}\times 1 \\ =\frac{1}{256} \end{gathered}$$

Therefore, the probability that the student will tick exactly five questions correctly is $\frac{1}{256}$.

Step 3: Calculate the probability that the student will tick at least four questions correctly

Thus, the probability that the student ticked at least $4$ questions correctly can be given by the sum of probabilities of student ticking exactly four questions and exactly five questions correctly

$$\begin{gathered} =\left(\frac{11}{256}+\frac{1}{256}\right) \\ =\frac{11+1}{256} \\ =\frac{12}{256} \\ =\frac{3}{64} \end{gathered}$$

Hence, the probability that the student will tick the correct option in at least four questions is $\frac{3}{64}$.