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An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (2008)
- a)2, 4 or 8
- b)3, 6 or 9
- c)4 or 8
- d)5 or 10
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
An experiment has 10 equally likely outcomes. Let A and B be non-empty...
We have n (S) = 10, n(A) = 4
Let n(B) = x and n(A ∩ B) = y
Then for A and B to be independent events P(A ∩ B) = P(A) P(B)
Let n(B) = x and n(A ∩ B) = y
Then for A and B to be independent events P(A ∩ B) = P(A) P(B)
⇒ y can be 2 or 4 so that x = 5 or 10
∴ n (B) = 5 or 10
∴ n (B) = 5 or 10
Most Upvoted Answer
An experiment has 10 equally likely outcomes. Let A and B be non-empty...
To understand why the correct answer is option 'D', let's break down the problem step by step:
Given:
- There are 10 equally likely outcomes in the experiment.
- Event A consists of 4 outcomes.
To find:
- The number of outcomes that event B must have so that A and B are independent.
Solution:
1. Independence of Events:
For two events A and B to be independent, the occurrence of event A should not affect the probability of event B, and vice versa. Mathematically, this can be represented as:
P(A ∩ B) = P(A) × P(B)
2. Total Outcomes:
We are given that there are 10 equally likely outcomes in the experiment. Therefore, the total number of outcomes is 10.
3. Event A:
Event A consists of 4 outcomes. Since there are 10 total outcomes, the probability of event A is:
P(A) = Number of outcomes in A / Total number of outcomes
= 4 / 10
= 2 / 5
4. Event B:
Let's assume that event B consists of 'x' outcomes. Since event A and event B are independent, the probability of event B should be unaffected by the occurrence of event A. Therefore, the probability of event B is:
P(B) = Number of outcomes in B / Total number of outcomes
= x / 10
5. Independence of Events:
According to the definition of independence of events, the probability of the intersection of events A and B should be equal to the product of their individual probabilities:
P(A ∩ B) = P(A) × P(B)
We can substitute the probabilities into this equation:
(4/10) × (x/10) = 2/5 × x/10
Simplifying this equation, we get:
4x = x
3x = 0
6. Conclusion:
From the equation above, we can see that the only way for the equation to hold true is if x = 0. This means that event B must have 0 outcomes. However, the problem states that event B must be non-empty. Therefore, it is not possible for event A and event B to be independent if event B has any outcomes.
Hence, the correct answer is option 'D': 5 or 10 outcomes.
Given:
- There are 10 equally likely outcomes in the experiment.
- Event A consists of 4 outcomes.
To find:
- The number of outcomes that event B must have so that A and B are independent.
Solution:
1. Independence of Events:
For two events A and B to be independent, the occurrence of event A should not affect the probability of event B, and vice versa. Mathematically, this can be represented as:
P(A ∩ B) = P(A) × P(B)
2. Total Outcomes:
We are given that there are 10 equally likely outcomes in the experiment. Therefore, the total number of outcomes is 10.
3. Event A:
Event A consists of 4 outcomes. Since there are 10 total outcomes, the probability of event A is:
P(A) = Number of outcomes in A / Total number of outcomes
= 4 / 10
= 2 / 5
4. Event B:
Let's assume that event B consists of 'x' outcomes. Since event A and event B are independent, the probability of event B should be unaffected by the occurrence of event A. Therefore, the probability of event B is:
P(B) = Number of outcomes in B / Total number of outcomes
= x / 10
5. Independence of Events:
According to the definition of independence of events, the probability of the intersection of events A and B should be equal to the product of their individual probabilities:
P(A ∩ B) = P(A) × P(B)
We can substitute the probabilities into this equation:
(4/10) × (x/10) = 2/5 × x/10
Simplifying this equation, we get:
4x = x
3x = 0
6. Conclusion:
From the equation above, we can see that the only way for the equation to hold true is if x = 0. This means that event B must have 0 outcomes. However, the problem states that event B must be non-empty. Therefore, it is not possible for event A and event B to be independent if event B has any outcomes.
Hence, the correct answer is option 'D': 5 or 10 outcomes.
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An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (2008)a)2, 4 or 8b)3, 6 or 9c)4 or 8d)5 or 10Correct answer is option 'D'. Can you explain this answer?
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An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (2008)a)2, 4 or 8b)3, 6 or 9c)4 or 8d)5 or 10Correct answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (2008)a)2, 4 or 8b)3, 6 or 9c)4 or 8d)5 or 10Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An experiment has 10 equally likely outcomes. Let A and B be non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is (2008)a)2, 4 or 8b)3, 6 or 9c)4 or 8d)5 or 10Correct answer is option 'D'. Can you explain this answer?.
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