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A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p, q and 1/2 respectively. If the probability that the student is successful is 1/2, then
- a)p = 1, q = 0
- b)p = 2/3, q = 1/2
- c)There are infinitely many values of p and q
- d)All of the above
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A student appears for test I, II and III. The student is successful if...
Let A, B and C be the events that the student is successful in test I, II and III respectively, then P(the student is successful)
P[A∩B∩C')∪(A∩B'∩C)∪(A∩B∩C)]
= P(A∩B∩C') + P(A∩B'∩C) + P(A∩B∩C)
P(A).P(B).P(C') + P(A)P(B')P(C) + P(A)P(B)P(C)
{∴ A, B, C are independent}
This equation has infinitely many values of p and q.
= P(A∩B∩C') + P(A∩B'∩C) + P(A∩B∩C)
P(A).P(B).P(C') + P(A)P(B')P(C) + P(A)P(B)P(C)
{∴ A, B, C are independent}
This equation has infinitely many values of p and q.
Most Upvoted Answer
A student appears for test I, II and III. The student is successful if...
To solve this problem, let's break it down step by step:
Step 1: Define the events:
Let A be the event of passing in Test I,
B be the event of passing in Test II, and
C be the event of passing in Test III.
Step 2: Determine the probability of success:
We are given that the probability of the student being successful is 1/2. Let S be the event of being successful. Therefore, P(S) = 1/2.
Step 3: Determine the given probabilities:
We are also given that the probabilities of passing in Test I, Test II, and Test III are p, q, and 1/2 respectively. Therefore, P(A) = p, P(B) = q, and P(C) = 1/2.
Step 4: Use conditional probability to calculate P(S):
We can express the event of being successful as the union of two events: passing in Test I and II (A∩B) or passing in Test I and III (A∩C). Therefore, we can write P(S) as follows:
P(S) = P(A∩B) + P(A∩C)
Step 5: Calculate the probabilities of passing in both combinations:
Using the multiplication rule of probability, we can calculate P(A∩B) and P(A∩C) as follows:
P(A∩B) = P(A) * P(B) = p * q
P(A∩C) = P(A) * P(C) = p * (1/2) = p/2
Step 6: Substitute the values into the equation for P(S):
Substituting the values we have calculated, we get:
1/2 = p * q + p/2
Step 7: Solve the equation:
Simplifying the equation, we have:
1/2 = (2pq + p) / 2
1 = 2pq + p
Step 8: Simplify the equation:
Rearranging the equation, we have:
2pq + p - 1 = 0
Step 9: Factor the equation:
Factoring out the common term 'p', we get:
p(2q + 1) - 1 = 0
Step 10: Solve for p:
Solving the equation for p, we get:
p = 1 / (2q + 1)
Step 11: Analyze the equation:
From the equation, we can see that for any value of q, there exists a corresponding value of p that satisfies the equation. Therefore, there are infinitely many values of p and q that satisfy the given conditions.
Conclusion: Hence, the correct answer is option 'D' - There are infinitely many values of p and q.
Step 1: Define the events:
Let A be the event of passing in Test I,
B be the event of passing in Test II, and
C be the event of passing in Test III.
Step 2: Determine the probability of success:
We are given that the probability of the student being successful is 1/2. Let S be the event of being successful. Therefore, P(S) = 1/2.
Step 3: Determine the given probabilities:
We are also given that the probabilities of passing in Test I, Test II, and Test III are p, q, and 1/2 respectively. Therefore, P(A) = p, P(B) = q, and P(C) = 1/2.
Step 4: Use conditional probability to calculate P(S):
We can express the event of being successful as the union of two events: passing in Test I and II (A∩B) or passing in Test I and III (A∩C). Therefore, we can write P(S) as follows:
P(S) = P(A∩B) + P(A∩C)
Step 5: Calculate the probabilities of passing in both combinations:
Using the multiplication rule of probability, we can calculate P(A∩B) and P(A∩C) as follows:
P(A∩B) = P(A) * P(B) = p * q
P(A∩C) = P(A) * P(C) = p * (1/2) = p/2
Step 6: Substitute the values into the equation for P(S):
Substituting the values we have calculated, we get:
1/2 = p * q + p/2
Step 7: Solve the equation:
Simplifying the equation, we have:
1/2 = (2pq + p) / 2
1 = 2pq + p
Step 8: Simplify the equation:
Rearranging the equation, we have:
2pq + p - 1 = 0
Step 9: Factor the equation:
Factoring out the common term 'p', we get:
p(2q + 1) - 1 = 0
Step 10: Solve for p:
Solving the equation for p, we get:
p = 1 / (2q + 1)
Step 11: Analyze the equation:
From the equation, we can see that for any value of q, there exists a corresponding value of p that satisfies the equation. Therefore, there are infinitely many values of p and q that satisfy the given conditions.
Conclusion: Hence, the correct answer is option 'D' - There are infinitely many values of p and q.
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A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer?
Question Description
A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer?.
A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A student appears for test I, II and III. The student is successful if he passes either in test I and II or test I and III. The probability of the student passing in test I, II, III are p,q and 1/2respectively. If the probability that the student is successful is 1/2,then a)p = 1, q = 0b)p = 2/3, q = 1/2c)There are infinitely many values of p and qd)All of the aboveCorrect answer is option 'D'. Can you explain this answer?.
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