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It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12 and chance of occurence of an event "only A" is 1/12 (where P(x) denote the probability of occurence of event 'x'). Value of 22P(B) equals
Correct answer is between '0.66,0.67'. Can you explain this answer?
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It is given that events A and B of an experiment are such that P(A / B...
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It is given that events A and B of an experiment are such that P(A / B...
To find: P(A) and P(B)
Solution:
We can use Bayes' theorem to find P(A) and P(B).
Bayes' theorem states that:
P(A / B) = P(B / A) x P(A) / P(B)
Substituting the given values, we get:
1 / 4 = (1 / 12) x P(A) / P(B)
Multiplying both sides by P(B), we get:
P(B) / 4 = P(A) / 12
Cross-multiplying, we get:
3P(B) = 4P(A)
Also, we know that:
P(A / B) x P(B) = P(A and B)
Substituting the given values, we get:
1 / 4 x P(B) = P(A and B)
Multiplying both sides by 4, we get:
P(B) = 4P(A and B)
Substituting this value in the equation 3P(B) = 4P(A), we get:
3(4P(A and B)) = 4P(A)
Simplifying, we get:
P(A and B) = 4P(A) / 12
P(A and B) = P(A) / 3
Now, we know that:
P(A and B) + P(A' and B) = P(B)
where A' is the complement of A.
Substituting the values, we get:
P(A) / 3 + P(B / A') x P(A') = P(B)
We are not given P(B / A'), but we do know that P(A and B') = 0, since events A and B are not independent.
Using the formula for conditional probability, we get:
P(A and B') = P(B') x P(A / B')
Since P(A and B') = 0, we get:
P(B') x P(A / B') = 0
This implies that either P(B') = 0 (which is not possible since the chance of occurrence of an event cannot be zero) or P(A / B') = 0.
If P(A / B') = 0, then events A and B' are independent, which contradicts the given information that events A and B are not independent.
Therefore, we can conclude that P(B / A') = 0.
Substituting this value in the equation above, we get:
P(A) / 3 + 0 x P(A') = P(B)
Simplifying, we get:
P(A) / 3 = P(B)
Substituting this value in the equation 3P(B) = 4P(A), we get:
3P(A) / 3 = 4P(A) / 4
Simplifying, we get:
P(A) = 3/7
Substituting this value in the equation P(A) / 3 = P(B), we get:
P(B) = 1/7
Therefore, the required probabilities are:
P(A) = 3/7
P(B) = 1/7
Solution:
We can use Bayes' theorem to find P(A) and P(B).
Bayes' theorem states that:
P(A / B) = P(B / A) x P(A) / P(B)
Substituting the given values, we get:
1 / 4 = (1 / 12) x P(A) / P(B)
Multiplying both sides by P(B), we get:
P(B) / 4 = P(A) / 12
Cross-multiplying, we get:
3P(B) = 4P(A)
Also, we know that:
P(A / B) x P(B) = P(A and B)
Substituting the given values, we get:
1 / 4 x P(B) = P(A and B)
Multiplying both sides by 4, we get:
P(B) = 4P(A and B)
Substituting this value in the equation 3P(B) = 4P(A), we get:
3(4P(A and B)) = 4P(A)
Simplifying, we get:
P(A and B) = 4P(A) / 12
P(A and B) = P(A) / 3
Now, we know that:
P(A and B) + P(A' and B) = P(B)
where A' is the complement of A.
Substituting the values, we get:
P(A) / 3 + P(B / A') x P(A') = P(B)
We are not given P(B / A'), but we do know that P(A and B') = 0, since events A and B are not independent.
Using the formula for conditional probability, we get:
P(A and B') = P(B') x P(A / B')
Since P(A and B') = 0, we get:
P(B') x P(A / B') = 0
This implies that either P(B') = 0 (which is not possible since the chance of occurrence of an event cannot be zero) or P(A / B') = 0.
If P(A / B') = 0, then events A and B' are independent, which contradicts the given information that events A and B are not independent.
Therefore, we can conclude that P(B / A') = 0.
Substituting this value in the equation above, we get:
P(A) / 3 + 0 x P(A') = P(B)
Simplifying, we get:
P(A) / 3 = P(B)
Substituting this value in the equation 3P(B) = 4P(A), we get:
3P(A) / 3 = 4P(A) / 4
Simplifying, we get:
P(A) = 3/7
Substituting this value in the equation P(A) / 3 = P(B), we get:
P(B) = 1/7
Therefore, the required probabilities are:
P(A) = 3/7
P(B) = 1/7
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It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. Can you explain this answer?
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It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. 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 It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. Can you explain this answer?.
It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. 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 It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for It is given that events A and B of an experiment are such that P(A / B) = 1 / 4 and P (B / A) = 1/12and chance of occurence of an event "only A" is 1/12(where P(x) denote the probability of occurence of event x). Value of 22P(B) equalsCorrect answer is between '0.66,0.67'. Can you explain this answer?.
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