With A,B,X defined as before, determine whether the following statements are true or false:
1. A and B are independent. (T/F)
2. A and B are conditionally independent, given X=0. (T/F)
3. A and B are conditionally independent, given X=1. (T/F)
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1. T: A and B are independent.
Why did the mathematician go to the circus? He wanted to find the probability of the clown bot telling a funny joke.
2. T: A and B are conditionally independent, given X=0.
Why did the clown become a mathematician? Because he wanted to clown-diagram events and find conditional probability.
3. F: A and B are not conditionally independent, given X=1.
Why did the probability theory book never get published? Because it had too many dependent events and no sense of conditional independence.
To determine if statements 1, 2, and 3 are true or false, we need to analyze the information provided about the variables A, B, and X.
However, you have not provided any information about these variables or their relationship, so we cannot determine the truth or falsehood of the statements without additional context or information.
Please provide more details about the variables A, B, and X and any relationships or conditions that exist between them.
To determine whether the statements are true or false, we need to understand the concept of independence and conditional independence.
1. For statement 1, "A and B are independent":
To check if A and B are independent, we need to understand whether the occurrence of A affects the probability of B and vice versa. In other words, we need to compare P(A∩B) with P(A) * P(B) to determine independence.
To do this, we need the values of P(A∩B), P(A), and P(B).
2. For statement 2, "A and B are conditionally independent, given X=0":
Here, we need to determine whether A and B are independent given the information that X=0. To do this, we need to compare P(A∩B | X=0) with P(A | X=0) * P(B | X=0).
To calculate these probabilities, we need the joint probabilities P(A∩B∩X=0), P(A∩X=0), and P(B∩X=0), as well as the conditional probabilities P(X=0), P(A | X=0), and P(B | X=0).
3. For statement 3, "A and B are conditionally independent, given X=1":
Similarly, to determine whether A and B are conditionally independent given X=1, we compare P(A∩B | X=1) with P(A | X=1) * P(B | X=1).
We need the joint probabilities P(A∩B∩X=1), P(A∩X=1), and P(B∩X=1), as well as the conditional probabilities P(X=1), P(A | X=1), and P(B | X=1).
By calculating these probabilities and comparing them, we can determine the truth or falsehood of each statement.