Question

There are 50 questions in an exam. A student’s probability of getting an answer correct is \[\dfrac{4}{5}\]. Then the probability that he is unable to correctly answer less than 2 questions
A. \[\dfrac{201}{5}{{(415)}^{50}}\]
B. \[201{{\left( \dfrac{1}{5} \right)}^{50}}\]
C. \[\dfrac{201}{5}{{\left( \dfrac{1}{5} \right)}^{50}}\]
D. \[\dfrac{201}{5}{{\left( \dfrac{4}{5} \right)}^{50}}\]

Answer 1

Hint: In the given problem a student’s probability of getting an answer correct is \[\dfrac{4}{5}\], we have to find the probability of getting incorrect answers. We have to find the probability that the student is unable to correctly answer less than 2 questions means that he may correctly answer 1 question out of 50 questions, or he may correctly answer 0 questions out of 50 questions. We will use the concept of binomial distribution, given by $ P\left( X \right)={}^{n}{{C}_{x}}{{p}^{x}}{{q}^{n-x}} $ , where for this question, we have X=N and N can be 0 or 1. Then, we have n=50, \[p=\dfrac{4}{5}\] and q=1-p.

Complete step-by-step answer:
We have been given the total number of questions in an exam = 50.
Also given that a student’s probability of getting an answer correct is \[p=\dfrac{4}{5}\]
Now we have to find the student’s probability of getting an answer incorrect. We know that the total sum of probability of favourable and unfavourable outcomes is 1. Therefore, we get
Probability of getting an answer incorrect is \[q=1-\dfrac{4}{5}=\dfrac{1}{5}\]
Let us suppose the student got a\[N\] number of questions correct out of 50 questions in the exam.
According to the problem given, we have to find a probability that he is unable to answer less than 2 correct answers. It means that either he gets 1 question out of 50 questions correct or he gets 0 questions out of 50 questions correct. So, we get the condition as \[N<2\].
We will apply the binomial distribution concept here, given by $ P\left( X \right)={}^{n}{{C}_{x}}{{p}^{x}}{{q}^{n-x}} $ , where n is the number of trials, X is the number of success desired, p is the probability of getting success in one trial and q is probability of getting failure in one trial.
We have X=N, n=50, \[p=\dfrac{4}{5}\] and \[q=\dfrac{1}{5}\].
The probability of getting less than 2 questions correct out of 50 questions is given by,
\[P(N<2)=P(N=0)+P(N=1)\]
Here, \[P(N=0)\] means probability of getting 0 answer correct and \[P(N=1)\] means probability of getting 1 answer correct out of 50 questions in an exam.
We can substitute the values in the formula and find the probability of getting 0 answer correct or all answers incorrect out of 50 questions is
\[\begin{align}
  & P(N=0)={}^{50}{{C}_{0}}{{\left( \dfrac{4}{5} \right)}^{0}}{{\left( \dfrac{1}{5} \right)}^{50-0}} \\
 & \Rightarrow P(N=0)=1\times 1\times {{\left( \dfrac{1}{5} \right)}^{50}} \\
 & \Rightarrow P(N=0)={{\left( \dfrac{1}{5} \right)}^{50}}.......\left( i \right) \\
\end{align}\]
And we get the probability of getting 1 answer correct out 50 questions as
\[\begin{align}
  & P\left( N=1 \right)={}^{50}{{C}_{1}}{{\left( \dfrac{4}{5} \right)}^{1}}{{\left( \dfrac{1}{5} \right)}^{50-1}} \\
 & \Rightarrow P\left( N=1 \right)={}^{50}{{C}_{1}}\left( \dfrac{4}{5} \right){{\left( \dfrac{1}{5} \right)}^{49}}.......\left( ii \right) \\
\end{align}\]
Now, we will add (i) and (ii) to find \[P(N<2)\]

\[P(N<2)=P(N=0)+P(N=1)\]
\[P(N<2)={{\left( \dfrac{1}{5} \right)}^{50}}+{}^{50}{{C}_{1}}\left( \dfrac{4}{5} \right){{\left( \dfrac{1}{5} \right)}^{49}}\]
\[\begin{align}
  & P(N<2)=\dfrac{1}{{{5}^{50}}}+50\times \dfrac{4}{5}\times \dfrac{1}{{{5}^{49}}} \\
 & \\
\end{align}\]
\[P(N<2)=\dfrac{1}{{{5}^{50}}}+\dfrac{50\times 4}{{{5}^{50}}}\]
\[P(N<2)=\dfrac{1+200}{{{5}^{50}}}\]
\[P(N<2)=\dfrac{201}{{{5}^{50}}}\]
\[P(N<2)=201{{\left( \dfrac{1}{5} \right)}^{50}}\]
Hence, we have got the probability that the student is unable to correctly answer less than 2 questions out of 50 questions in the exam as \[201{{\left( \dfrac{1}{5} \right)}^{50}}\].
Option B is the correct one.

Note: Students must realise that this question required binomial distribution to be applied. They must find out the necessary parameters like probability of getting an answer incorrect that helps in calculating the required answer. The next point is regarding the condition of P (N < 2). Sometimes, students consider it as \[P\left( N\le 2 \right)\] , but here in the question it is asked to find probability for less than 2 correct answers and not including 2. So, students must read the question carefully, then form the right conditions.