# Probability Q: Show A & B Independent

• GreenPrint
In summary, the problem is to show that A and B are independent given the information that P(A|B) = P(A|B^c), and that P(B) and P(B^c) are both greater than 0. The definition of independence is that P(A intersection B) = P(A)P(B). To solve this problem, we can use the fact that P(A)= P(A|B)P(B)+ P(A|B^c)B^c, for any A and B, and that P(B^c)= 1- P(B).
GreenPrint

## Homework Statement

Suppose
P(A|B) = P(A|B^c)
(P(B) > 0 and P(B^c) > 0 both understood). Show that A and B are independent.

## The Attempt at a Solution

I don't know where to go from here. Thanks for any help

$\frac{P(A \bigcap B)}{P(B)} = \frac{P(A \bigcap B^{c})}{P(B^{c})}$

Last edited:
Please restate (in fact, go back are reread) the problem. As you have it now it says "Suppose A and B are independent. Show that A and B are independent"!

Ok sorry about that I fixed that. Thanks for helping me.

Okay, good. Now what is your definition of "independent events"?

P(A intersection B) = P(A)P(B)

not sure what to do from here because it's conditional probabilities.

Good! So you are given that $P(A|B)= P(A|B^c)$ and want to prove that $P(A\cap B)= P(A)P(B)$.

You should know, then, that $P(A)= P(A|B)P(B)+ P(A|B^c)B^c$, for any A and B. Further, $P(B^c)= 1- P(B)$.

(The reason I asked about the definition was that some texts use "P(A)= P(A|B)" as the definition of "A and B are independent. Of course it is easy to show that is equivalent to your "$P(A\cap B)= P(A)P(B)$".)

## 1. What does it mean for events A and B to be independent?

Two events A and B are independent if the occurrence of one does not affect the probability of the other occurring.

## 2. How do you calculate the probability of two independent events occurring together?

To calculate the probability of two independent events A and B occurring together, simply multiply the individual probabilities of A and B. This can be represented as P(A and B) = P(A) x P(B).

## 3. Can two events be considered independent if they have a correlation?

No, two events cannot be considered independent if they have a correlation. Independence means that the occurrence of one event does not affect the probability of the other occurring, whereas correlation implies a relationship between the two events.

## 4. How can you determine if two events are independent using a Venn diagram?

If two events A and B are independent, then their Venn diagram will show that the two circles representing A and B do not overlap. This indicates that the occurrence of one event does not affect the probability of the other event occurring.

## 5. Can the probability of an event A change if it is independent from another event B?

No, the probability of an event A will not change if it is independent from another event B. This is because the probability of A is not affected by the occurrence or non-occurrence of event B.

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