Question:
A multiple choice test consists of \(5\) questions, each question having \(4\) responses. There is only one correct response and the remaining \(3\) are incorrect responses. If a candidate attempts all the \(5\) questions, then the probability that he answers at least \(3\) questions incorrectly is
A multiple choice test consists of \(5\) questions, each question having \(4\) responses. There is only one correct response and the remaining \(3\) are incorrect responses. If a candidate attempts all the \(5\) questions, then the probability that he answers at least \(3\) questions incorrectly is
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For MCQ guessing problems, use binomial distribution where the probability of success or failure remains the same for each question.
\[
P(X=r)={}^nC_rp^rq^{n-r}
\]
Updated On: Jun 25, 2026
- \(\dfrac{675}{1024}\)
- \(\dfrac{459}{512}\)
- \(\dfrac{81}{128}\)
- \(\dfrac{135}{512}\)
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The Correct Option is B
Solution and Explanation
Step 1: Identify the probability of incorrect response.
Each question has \(4\) responses, out of which \(1\) is correct and \(3\) are incorrect.
Therefore, probability of an incorrect response is \[ p=\frac{3}{4} \] Probability of a correct response is \[ q=\frac{1}{4} \] Total number of questions is \[ n=5 \]
Step 2: Define the random variable.
Let \(X\) be the number of incorrectly answered questions.
Then, \[ X\sim B\left(5,\frac{3}{4}\right) \] We need \[ P(X\geq 3) \] So, \[ P(X\geq 3)=P(X=3)+P(X=4)+P(X=5) \]
Step 3: Apply binomial probability formula.
Using \[ P(X=r)={}^nC_r p^r q^{n-r}, \] we get \[ P(X\geq 3) = {}^5C_3\left(\frac{3}{4}\right)^3\left(\frac{1}{4}\right)^2 + {}^5C_4\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right)^1 + {}^5C_5\left(\frac{3}{4}\right)^5 \]
Step 4: Simplify each term.
First term: \[ {}^5C_3\left(\frac{3}{4}\right)^3\left(\frac{1}{4}\right)^2 = 10\cdot \frac{27}{64}\cdot \frac{1}{16} \] \[ = \frac{270}{1024} \] Second term: \[ {}^5C_4\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right) = 5\cdot \frac{81}{256}\cdot \frac{1}{4} \] \[ = \frac{405}{1024} \] Third term: \[ {}^5C_5\left(\frac{3}{4}\right)^5 = 1\cdot \frac{243}{1024} \] \[ = \frac{243}{1024} \]
Step 5: Add the probabilities.
Therefore, \[ P(X\geq 3) = \frac{270}{1024} + \frac{405}{1024} + \frac{243}{1024} \] \[ = \frac{918}{1024} \] Simplifying, \[ \frac{918}{1024} = \frac{459}{512} \]
Step 6: Final conclusion.
Hence, the required probability is \[ \boxed{\frac{459}{512}} \]
Each question has \(4\) responses, out of which \(1\) is correct and \(3\) are incorrect.
Therefore, probability of an incorrect response is \[ p=\frac{3}{4} \] Probability of a correct response is \[ q=\frac{1}{4} \] Total number of questions is \[ n=5 \]
Step 2: Define the random variable.
Let \(X\) be the number of incorrectly answered questions.
Then, \[ X\sim B\left(5,\frac{3}{4}\right) \] We need \[ P(X\geq 3) \] So, \[ P(X\geq 3)=P(X=3)+P(X=4)+P(X=5) \]
Step 3: Apply binomial probability formula.
Using \[ P(X=r)={}^nC_r p^r q^{n-r}, \] we get \[ P(X\geq 3) = {}^5C_3\left(\frac{3}{4}\right)^3\left(\frac{1}{4}\right)^2 + {}^5C_4\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right)^1 + {}^5C_5\left(\frac{3}{4}\right)^5 \]
Step 4: Simplify each term.
First term: \[ {}^5C_3\left(\frac{3}{4}\right)^3\left(\frac{1}{4}\right)^2 = 10\cdot \frac{27}{64}\cdot \frac{1}{16} \] \[ = \frac{270}{1024} \] Second term: \[ {}^5C_4\left(\frac{3}{4}\right)^4\left(\frac{1}{4}\right) = 5\cdot \frac{81}{256}\cdot \frac{1}{4} \] \[ = \frac{405}{1024} \] Third term: \[ {}^5C_5\left(\frac{3}{4}\right)^5 = 1\cdot \frac{243}{1024} \] \[ = \frac{243}{1024} \]
Step 5: Add the probabilities.
Therefore, \[ P(X\geq 3) = \frac{270}{1024} + \frac{405}{1024} + \frac{243}{1024} \] \[ = \frac{918}{1024} \] Simplifying, \[ \frac{918}{1024} = \frac{459}{512} \]
Step 6: Final conclusion.
Hence, the required probability is \[ \boxed{\frac{459}{512}} \]
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