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The problem shown on the image is asking about the conditions for two events, \(A\) and \(B\), to be considered independent. Specifically, it asks which pairs of probabilities need to be equal for the events to be independent.

For two events to be independent, the condition is:
\[
P(A \cap B) = P(A) \cdot P(B)
\]
This means that the probability of both events \(A\) and \(B\) happening together is equal to the product of their individual probabilities.

Let's now look at the provided options and analyze them:
1. **\(P(B) = P(B | A)\):** This is correct. For events \(A\) and \(B\) to be independent, the probability of \(B\) happening given \(A\) (conditional probability) must be the same as the probability of \(B\) happening regardless of \(A\). This holds when \(A\) and \(B\) are independent.
2. **\(P(A | B) = P(B | A)\):** This is incorrect. Conditional probabilities \(P(A | B)\) and \(P(B | A)\) are not necessarily equal, even for independent events.
3. **\(P(A) = P(A \cap B)\):** This is incorrect. \(P(A)\) should not equal \(P(A \cap B)\) unless the event \(B\) always occurs when \(A\) does, which is not a condition for independence.
4. **\(P(A) = P(B)\):** This is incorrect. The probabilities of \(A\) and \(B\) do not need to be equal for independence.
5. **\(P(A) = P(A | B)\):** This is correct. The probability of \(A\) happening, given that \(B\) occurred, should be the same as the unconditional probability of \(A\) for independent events.

### The correct answers are:
- \(P(B) = P(B | A)\)
- \(P(A) = P(A | B)\)

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the formula for conditional probability?
2. How do you prove that two events are independent?
3. What is the difference between mutually exclusive and independent events?
4. Can two events with zero probability be independent?
5. How does the complement of an event affect its independence?

**Tip:** Always remember that independence means one event does not affect the probability of the other event happening.

If we have two events A and B, for them to be considered independent, which pair of probabilities need to be equal to each other? Select all that apply.

Understand the conditions for two events, A and B, to be independent. This problem explains which probabilities need to be equal, such as P(B) = P(B | A) and P(A) = P(A | B). A detailed explanation of independence theorem in probability theory.

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