Probability Q: Show A & B Independent

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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).
  • #1
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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.

Homework Equations


The Attempt at a Solution



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

[itex]\frac{P(A \bigcap B)}{P(B)} = \frac{P(A \bigcap B^{c})}{P(B^{c})}[/itex]
 
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  • #2
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"!
 
  • #3
Ok sorry about that I fixed that. Thanks for helping me.
 
  • #4
Okay, good. Now what is your definition of "independent events"?
 
  • #5
P(A intersection B) = P(A)P(B)

not sure what to do from here because it's conditional probabilities.
 
  • #6
Good! So you are given that [itex]P(A|B)= P(A|B^c)[/itex] and want to prove that [itex]P(A\cap B)= P(A)P(B)[/itex].

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

(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 "[itex]P(A\cap
B)= P(A)P(B)[/itex]".)
 

Related to Probability Q: Show A & B Independent

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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