- #1

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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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- #1

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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?

- #2

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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.

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?

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

Dale

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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.If A and B are independent then the probability of both happening at once should be 0.

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.

- #4

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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.

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?

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.

- #5

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Although, as it turned out, one goal from Sanchez was enough!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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- #7

Dale

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Why not? I usually consider it in terms of Venn diagrams also where the area of the region is the probability.n 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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Because it can mislead you to a wrong conclusion as in post #1.Why not?

- #9

Dale

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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.Because it can mislead you to a wrong conclusion as in post #1.

- #10

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So how to correctly draw the Venn diagram in this case?But that is because he drew the Venn diagram wrong, not because a correctly drawn Venn diagram is not useful in probability.

- #11

Dale

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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.So how to correctly draw the Venn diagram in this case?

- #12

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How such a diagram would tell us that A and B are independent?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.

- #13

Dale

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You have it backwards. We were given that A and B were independent, and drew the diagram accordingly.How such a diagram would tell us that A and B are independent?

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

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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?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.

- #15

Dale

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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.pptDo you know any reference where such diagrams are really used in practice?

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.

- #16

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It's the way I remember Bayes' theorem.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?

##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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