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The probability of \(A\) given \(B\), denoted as \(P(A|B)\), is the probability of event \(A\) happening when we know that event \(B\) has happened. On the other hand, the probability of \(B\) given \(A\), denoted as \(P(B|A)\), is the probability of event \(B\) happening when we know that event \(A\) has happened.

Let's consider a scenario where we have a deck of playing cards. Suppose event \(A\) is getting a king and event \(B\) is getting a red card. We want to find the probabilities of \(A\) given \(B\) and \(B\) given \(A\). - \(P(A|B)\) There are 26 red cards in a deck of 52 playing cards. Out of these 26 red cards, only 2 are kings. Therefore, the probability of getting a king, given that we have a red card, is: \[P(A|B) = \frac{2}{26} = \frac{1}{13}\] - \(P(B|A)\) There are 4 kings in a deck of 52 playing cards. Out of these 4 kings, 2 are red cards. Therefore, the probability of getting a red card, given that we have a king, is: \[P(B|A) = \frac{2}{4} = \frac{1}{2}\] Since \(\frac{1}{13} \neq \frac{1}{2}\), we can conclude that \(P(A|B)\) is not the same as \(P(B|A)\).

We can also show the formulas for each conditional probability, which are as follows: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\] \[P(B|A) = \frac{P(A \cap B)}{P(A)}\] As we can see, the denominators for each formula are different (\(P(B)\) and \(P(A)\)), which indicates that in general, the two conditional probabilities are not the same. This further supports our argument that the probability of \(A\) given \(B\) is not the same as the probability of \(B\) given \(A\).