Question:
In a multiple-choice examination, there are \(10\) questions with one correct option out of \(4\) options for each question. A student gets \(4\) marks for each correct answer and \(1\) mark is deducted for each incorrect answer. The probability that a student, guessing randomly on every question, scores \(30\) marks in this exam is...
In a multiple-choice examination, there are \(10\) questions with one correct option out of \(4\) options for each question. A student gets \(4\) marks for each correct answer and \(1\) mark is deducted for each incorrect answer. The probability that a student, guessing randomly on every question, scores \(30\) marks in this exam is...
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Find the number of correct answers from \(4k-(10-k)=30\), then use the binomial formula.
Updated On: Oct 1, 2026
- \(\frac{45}{4^{10}}\)
- \(\frac{135}{4^{10}}\)
- \(\frac{405}{4^{10}}\)
- \(\frac{810}{4^{10}}\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
Let \(k\) be the number of correct answers. The other \(10-k\) are wrong. The score is \(4k - (10-k) = 5k - 10\).
Step 2: Find k:
\(5k - 10 = 30\), so \(k = 8\).
Step 3: Binomial probability:
Each guess is right with probability \(p = \frac14\) and wrong with \(q = \frac34\).
\[ P(X=8) = \binom{10}{8}\left(\frac14\right)^8\left(\frac34\right)^2 = 45\cdot\frac{9}{4^{10}} = \frac{405}{4^{10}} \]
Final Answer:
The probability is \(\frac{405}{4^{10}}\), option (C). \[ \boxed{\frac{405}{4^{10}}} \]
Let \(k\) be the number of correct answers. The other \(10-k\) are wrong. The score is \(4k - (10-k) = 5k - 10\).
Step 2: Find k:
\(5k - 10 = 30\), so \(k = 8\).
Step 3: Binomial probability:
Each guess is right with probability \(p = \frac14\) and wrong with \(q = \frac34\).
\[ P(X=8) = \binom{10}{8}\left(\frac14\right)^8\left(\frac34\right)^2 = 45\cdot\frac{9}{4^{10}} = \frac{405}{4^{10}} \]
Final Answer:
The probability is \(\frac{405}{4^{10}}\), option (C). \[ \boxed{\frac{405}{4^{10}}} \]
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