How is P(A∩B) = P(A)P(B) ?

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If A and B are independent then the probability of both happening at once should be 0.
If we drew a ven diagram it'd be just two circles who don't intersect.

I'm guessing I got wrong definition of independent, can someone explain please?
 

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  • #2
QuantumQuest
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If A and B are independent then the probability of both happening at once should be 0.
If we drew a ven diagram it'd be just two circles who don't intersect.

I'm guessing I got wrong definition of independent, can someone explain please?
By definition, two events ##A## and ##B## are independent if the occurrence of one does not affect the probability of occurrence of the other. The concept of independence extends to more than two events, taking pairwise independence for every pair of events inside a finite set of events.

In order to understand why ##P(A\cap B) = P(A)P(B)## you can write this expression using conditional probabilities

##P(A) = \frac{P(A)P(B)}{P(B)} = \frac{P(A\cap B)}{P(B)} = P(A | B)## and similarly for ##P(B)##.

So, you can see in a more intuitive manner that the occurrence of one event does not affect the probability of occurrence of the other.
 
  • #3
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If A and B are independent then the probability of both happening at once should be 0.
This is "mutually exclusive", not "independent". Think about it in terms of wagers. Suppose you are betting that it will rain tomorrow. Depending on your location you might accept 10:1 odds on such a wager.

Now, suppose that you also know that tomorrow is Tuesday, would that change the odds you would accept? If not, then they are independent.

If two events are mutually exclusive then knowing one definitely changes the wager you would accept on the other. So they are not independent.
 
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PeroK
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If A and B are independent then the probability of both happening at once should be 0.
If we drew a ven diagram it'd be just two circles who don't intersect.

I'm guessing I got wrong definition of independent, can someone explain please?
If event A is Chelsea beating Spurs in the FA cup and event B is Arsenal beating Man City tomorrow, then those are independent, but they could both happen.

But, if event B was Spurs beating Chelsea, then events A and B are not independent, but mutually exclusive.

An example of two events that are dependent are event A that Arsenal win and event B that Sanchez scores a hat trick.
 
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PeroK
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An example of two events that are dependent are event A that Arsenal win and event B that Sanchez scores a hat trick.
Although, as it turned out, one goal from Sanchez was enough!
 
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Demystifier
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The most confusing thing here is the symbol ##\cap##. The meaning of this symbol in set theory is very different from the meaning of the same symbol in probability theory. In set theory it can be interpreted in terms of Venn diagrams, but such an interpretation is not very useful in probability theory.
 
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n set theory it can be interpreted in terms of Venn diagrams, but such an interpretation is not very useful in probability theory.
Why not? I usually consider it in terms of Venn diagrams also where the area of the region is the probability.
 
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Because it can mislead you to a wrong conclusion as in post #1.
But that is because he drew the Venn diagram wrong, not because a correctly drawn Venn diagram is not useful in probability.
 
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Demystifier
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But that is because he drew the Venn diagram wrong, not because a correctly drawn Venn diagram is not useful in probability.
So how to correctly draw the Venn diagram in this case?
 
  • #11
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So how to correctly draw the Venn diagram in this case?
Draw a square of area 1, a circle of area P(A) and a circle of area P(B). Position them such that both circles are inside the square and their overlap has area P(A∩B). The shape of the circles can be distorted if needed.
 
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Draw a square of area 1, a circle of area P(A) and a circle of area P(B). Position them such that both circles are inside the square and their overlap has area P(A∩B). The shape of the circles can be distorted if needed.
How such a diagram would tell us that A and B are independent?
 
  • #13
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How such a diagram would tell us that A and B are independent?
You have it backwards. We were given that A and B were independent, and drew the diagram accordingly.

If instead we were given a Venn diagram where the area represents probability then you can simply check if P(A∩B) = P(A) P(B) by looking at the corresponding areas.
 
  • #14
Demystifier
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You have it backwards. We were given that A and B were independent, and drew the diagram accordingly.

If instead we were given a Venn diagram where the area represents probability then you can simply check if P(A∩B) = P(A) P(B) by looking at the corresponding areas.
OK, you can do it, but in my opinion it's not very useful. Do you know any reference where such diagrams are really used in practice?
 
  • #15
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Do you know any reference where such diagrams are really used in practice?
Such diagrams were used to explain conditional probability and Bayes theorem to me when I was a student. Here is a lecture that takes the same approach: http://math.arizona.edu/~sreyes/math115as08/S08Proj1-BayesThm.ppt

I have not done a survey, but I have the impression that it is a common pedagogical technique. Certainly I would guess that the OP's teacher took that approach.
 
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PeroK
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OK, you can do it, but in my opinion it's not very useful. Do you know any reference where such diagrams are really used in practice?
It's the way I remember Bayes' theorem.

##P(A|B) = \frac{P(B|A)P(A)}{P(B)}##

For some reason this is something I always find hard to remember. So, I draw a Venn diagram of two overlapping sets ##A, B## and note that:

##P(A|B) = \frac{P(A \cap B)}{P(B)}##

That's just the area of ##A \cap B## divided by the area of ##B##.

Likewise:

##P(B|A) = \frac{P(A \cap B)}{P(A)}##

And then I put the two together.

One day, perhaps, I will memorise Bayes' Theorem directly!
 
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Demystifier
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I still find the Venn diagrams in probability calculus more confusing than useful. But perhaps that's just me.
 
  • #18
FactChecker
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Here are two example Venn diagrams of independent and dependent events. But I only use it to visualize the concept and don't know if it is a practical method for determining independence.
independentEvents.png
 

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