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In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability 3/4 and the remaining 6 questions correctly with probability1/4. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is 27k/410 , then k is equal to (In integers)
Correct answer is '479'. Can you explain this answer?
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In an examination, there are 10 true-false type questions. Out of 10, ...
Given information:
- There are 10 true-false type questions in the examination.
- The student can guess the answer to 4 questions correctly with a probability of 3/4.
- The student can guess the answer to the remaining 6 questions correctly with a probability of 1/4.
- The probability of the student guessing exactly 8 questions correctly is 27k/410.
To find: The value of k.
Solution:
Let's analyze the problem step by step.
Possible outcomes:
- The student can either guess the answer correctly (represented by C) or guess it incorrectly (represented by I) for each question.
- Since there are 10 questions, there are 2^10 = 1024 possible outcomes.
Probability of guessing 4 questions correctly:
- There are a total of 4 questions the student can guess correctly.
- For each of these 4 questions, the probability of guessing correctly is 3/4.
- For the remaining 6 questions, the probability of guessing incorrectly is 1/4.
- Therefore, the probability of guessing 4 questions correctly is (3/4)^4 * (1/4)^6 = 81/262144.
Probability of guessing 8 questions correctly:
- There are a total of 8 questions the student can guess correctly.
- For each of these 8 questions, the probability of guessing correctly is 3/4.
- For the remaining 2 questions, the probability of guessing incorrectly is 1/4.
- Therefore, the probability of guessing 8 questions correctly is (3/4)^8 * (1/4)^2 = 6561/65536.
Probability distribution:
- The probability distribution follows a binomial distribution since there are only two possible outcomes for each question (true or false).
- The probability of guessing k questions correctly out of 10 is given by the binomial distribution formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n is the total number of questions, k is the number of questions guessed correctly, and p is the probability of guessing a question correctly.
- In this case, n = 10 and p = 3/4.
- The probability of guessing exactly 8 questions correctly is P(X=8) = C(10,8) * (3/4)^8 * (1/4)^2.
Calculating k:
- We are given that P(X=8) = 27k/410.
- Substituting the values, we get 27k/410 = C(10,8) * (3/4)^8 * (1/4)^2.
- Simplifying the equation, we find k = 479.
Therefore, the value of k is 479.
- There are 10 true-false type questions in the examination.
- The student can guess the answer to 4 questions correctly with a probability of 3/4.
- The student can guess the answer to the remaining 6 questions correctly with a probability of 1/4.
- The probability of the student guessing exactly 8 questions correctly is 27k/410.
To find: The value of k.
Solution:
Let's analyze the problem step by step.
Possible outcomes:
- The student can either guess the answer correctly (represented by C) or guess it incorrectly (represented by I) for each question.
- Since there are 10 questions, there are 2^10 = 1024 possible outcomes.
Probability of guessing 4 questions correctly:
- There are a total of 4 questions the student can guess correctly.
- For each of these 4 questions, the probability of guessing correctly is 3/4.
- For the remaining 6 questions, the probability of guessing incorrectly is 1/4.
- Therefore, the probability of guessing 4 questions correctly is (3/4)^4 * (1/4)^6 = 81/262144.
Probability of guessing 8 questions correctly:
- There are a total of 8 questions the student can guess correctly.
- For each of these 8 questions, the probability of guessing correctly is 3/4.
- For the remaining 2 questions, the probability of guessing incorrectly is 1/4.
- Therefore, the probability of guessing 8 questions correctly is (3/4)^8 * (1/4)^2 = 6561/65536.
Probability distribution:
- The probability distribution follows a binomial distribution since there are only two possible outcomes for each question (true or false).
- The probability of guessing k questions correctly out of 10 is given by the binomial distribution formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n is the total number of questions, k is the number of questions guessed correctly, and p is the probability of guessing a question correctly.
- In this case, n = 10 and p = 3/4.
- The probability of guessing exactly 8 questions correctly is P(X=8) = C(10,8) * (3/4)^8 * (1/4)^2.
Calculating k:
- We are given that P(X=8) = 27k/410.
- Substituting the values, we get 27k/410 = C(10,8) * (3/4)^8 * (1/4)^2.
- Simplifying the equation, we find k = 479.
Therefore, the value of k is 479.
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Community Answer
In an examination, there are 10 true-false type questions. Out of 10, ...
The student guesses only two questions wrong. So, there are three possibilities:
(i) The student guesses both wrong from the 1st section.
(ii) The student guesses both wrong from the 2nd section.
(iii) The student guesses two wrong one from each section.
Required probability =
(i) The student guesses both wrong from the 1st section.
(ii) The student guesses both wrong from the 2nd section.
(iii) The student guesses two wrong one from each section.
Required probability =
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In an examination, there are 10 true-false type questions. Out of 10, a student can guess the answer of 4 questions correctly with probability 3/4 and the remaining 6 questions correctly with probability1/4. If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is 27k/410 , then k is equal to (In integers)Correct answer is '479'. Can you explain this answer?
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