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An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches – 0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained all these students is
- a)0
- b)2550
- c)7525
- d)9375
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
An examination paper has 150 multiple-choice questions of one mark eac...
Ans.
Method to Solve :
Probability of choosing the correct option = 1/4
Probability of choosing a wrong option = 3/4
So, expected mark for a question for a student = 1/4�1+3/4�(−0.25)=1/16
Expected mark for a student for 150questions =1/16�150=9.375
So, sum total of the expected marks obtained by all 1000students = 9.375�1000=9375
Probability of choosing a wrong option = 3/4
So, expected mark for a question for a student = 1/4�1+3/4�(−0.25)=1/16
Expected mark for a student for 150questions =1/16�150=9.375
So, sum total of the expected marks obtained by all 1000students = 9.375�1000=9375
Most Upvoted Answer
An examination paper has 150 multiple-choice questions of one mark eac...
Calculation of Expected Marks
To solve this problem, we need to calculate the expected marks obtained by all the students who randomly answer the multiple-choice questions. Let's break down the steps for the calculation:
Step 1: Probability of Correct Answer
Each question has four choices, so the probability of choosing the correct answer is 1/4.
Step 2: Probability of Incorrect Answer
Since there are three incorrect choices for each question, the probability of choosing an incorrect answer is 3/4.
Step 3: Expected Mark for Correct Answer
For each correct answer, the student receives 1 mark.
Step 4: Expected Mark for Incorrect Answer
For each incorrect answer, the student loses 0.25 marks.
Step 5: Expected Mark Calculation
To calculate the expected mark, we multiply the probability of each outcome (correct or incorrect) by the corresponding mark value and sum them up.
Calculation:
The expected mark for a single question can be calculated as follows:
(Expected Mark for Correct Answer) * (Probability of Correct Answer) + (Expected Mark for Incorrect Answer) * (Probability of Incorrect Answer)
= (1 * 1/4) + (-0.25 * 3/4)
= 0.25 - 0.1875
= 0.0625
Therefore, the expected mark for a single question is 0.0625.
Total Expected Marks:
Since there are 150 questions in the examination paper, we can calculate the total expected marks by multiplying the expected mark for a single question by the number of questions.
Total Expected Marks = (Expected Mark for a Single Question) * (Number of Questions)
= 0.0625 * 150
= 9.375
Conclusion:
The sum total of the expected marks obtained by all the students who choose their answers randomly is 9.375. However, since the marks are given in integers, the answer is rounded off to the nearest integer, which is 9. Therefore, the correct option is (D) 9375.
To solve this problem, we need to calculate the expected marks obtained by all the students who randomly answer the multiple-choice questions. Let's break down the steps for the calculation:
Step 1: Probability of Correct Answer
Each question has four choices, so the probability of choosing the correct answer is 1/4.
Step 2: Probability of Incorrect Answer
Since there are three incorrect choices for each question, the probability of choosing an incorrect answer is 3/4.
Step 3: Expected Mark for Correct Answer
For each correct answer, the student receives 1 mark.
Step 4: Expected Mark for Incorrect Answer
For each incorrect answer, the student loses 0.25 marks.
Step 5: Expected Mark Calculation
To calculate the expected mark, we multiply the probability of each outcome (correct or incorrect) by the corresponding mark value and sum them up.
Calculation:
The expected mark for a single question can be calculated as follows:
(Expected Mark for Correct Answer) * (Probability of Correct Answer) + (Expected Mark for Incorrect Answer) * (Probability of Incorrect Answer)
= (1 * 1/4) + (-0.25 * 3/4)
= 0.25 - 0.1875
= 0.0625
Therefore, the expected mark for a single question is 0.0625.
Total Expected Marks:
Since there are 150 questions in the examination paper, we can calculate the total expected marks by multiplying the expected mark for a single question by the number of questions.
Total Expected Marks = (Expected Mark for a Single Question) * (Number of Questions)
= 0.0625 * 150
= 9.375
Conclusion:
The sum total of the expected marks obtained by all the students who choose their answers randomly is 9.375. However, since the marks are given in integers, the answer is rounded off to the nearest integer, which is 9. Therefore, the correct option is (D) 9375.
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An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches – 0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained all these students is a)0 b)2550 c)7525 d)9375 Correct answer is option 'D'. Can you explain this answer?
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