chopasticks said:how can i proof that >>>>>> p(A | B') = p(A | B)
p(A|B) is a conditional probability that measures the likelihood of event A occurring given that event B has already occurred. It is typically represented as p(A|B) = P(A and B) / P(B).
This statement means that the conditional probability of event A occurring given that event B has occurred is equal to the conditional probability of event A occurring given that event B has not occurred. In other words, the outcome of event B has no effect on the likelihood of event A occurring.
This notation indicates that events A and B are statistically independent, meaning that the occurrence of one event does not affect the likelihood of the other event occurring. In other words, the probability of event A remains the same regardless of the occurrence of event B.
There are multiple ways to prove this equality, but one method is to use the definition of conditional probability and basic algebra. First, we can express p(A|B) and p(A|B') in terms of joint probabilities, P(A and B) and P(A and B'), respectively. Then, we can manipulate the equations to show that they are equal. For a step-by-step guide, you can refer to the article "Proving p(A|B)=p(A|B'): A Step-By-Step Guide."
This relationship is important in understanding the potential causal relationship between two events. If p(A|B) = p(A|B'), it suggests that event B does not have a direct influence on event A, and any observed correlation between the two events may be due to chance rather than a causal relationship. This is a crucial consideration in many scientific studies and experiments.