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elmo
elmo
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I believe the idea of independence is, using your Venn diagram, that the proportion of A in B inis equal to the proportion of A in Ω. That is, AB takes up half the space of B, exactly as much as A takes out of Ω.

If we now interpret the Venn diagram in terms of probabilities, this means that given that B has occurred, A has ½ of occurring (=0.25/0.5), which is exactly the probability of A occurring anyways (=0.5/1), therefore knowledge/occurrence of B does not change the probability of A.

If you add a fifth event assigning now each outcome a probability of 0.2, then those proportions are not equal anymore.

We can generalize from your illustrative example and this geometric intuition to define independence as equal proportions, P(A|B) = P(A), that is, the probability of A happening (in general) equals the the probability of it happening (specifically) when B has happened, just as in your Venn diagram. Now P(A|B) = P(AB) / P(B), which is the proportion that A occupies in B. Substituting P(A) for P(A|B), we get the formal definition of independence for two events: P(AB) = P(A)P(B)

I believe the idea of independence is, using your Venn diagram, that the proportion of A in B in equal to the proportion of A in Ω. That is, AB takes up half the space of B, exactly as much as A takes out of Ω.

If we now interpret the Venn diagram in terms of probabilities, this means that given that B has occurred, A has ½ of occurring (=0.25/0.5), which is exactly the probability of A occurring anyways (=0.5/1), therefore knowledge/occurrence of B does not change the probability of A.

If you add a fifth event assigning now each outcome a probability of 0.2, then those proportions are not equal anymore.

We can generalize from your illustrative example and this geometric intuition to define independence as equal proportions, P(A|B) = P(A), that is, the probability of A happening (in general) equals the the probability of it happening (specifically) when B has happened, just as in your Venn diagram. Now P(A|B) = P(AB) / P(B), which is the proportion that A occupies in B. Substituting P(A) for P(A|B), we get the formal definition of independence for two events: P(AB) = P(A)P(B)

I believe the idea of independence is, using your Venn diagram, that the proportion of A in B is equal to the proportion of A in Ω. That is, AB takes up half the space of B, exactly as much as A takes out of Ω.

If we now interpret the Venn diagram in terms of probabilities, this means that given that B has occurred, A has ½ of occurring (=0.25/0.5), which is exactly the probability of A occurring anyways (=0.5/1), therefore knowledge/occurrence of B does not change the probability of A.

If you add a fifth event assigning now each outcome a probability of 0.2, then those proportions are not equal anymore.

We can generalize from your illustrative example and this geometric intuition to define independence as equal proportions, P(A|B) = P(A), that is, the probability of A happening (in general) equals the the probability of it happening (specifically) when B has happened, just as in your Venn diagram. Now P(A|B) = P(AB) / P(B), which is the proportion that A occupies in B. Substituting P(A) for P(A|B), we get the formal definition of independence for two events: P(AB) = P(A)P(B)

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elmo
elmo
  • 1
  • 2

I believe the idea of independence is, using your Venn diagram, that the proportion of A in B in equal to the proportion of A in Ω. That is, AB takes up half the space of B, exactly as much as A takes out of Ω.

If we now interpret the Venn diagram in terms of probabilities, this means that given that B has occurred, A has ½ of occurring (=0.25/0.5), which is exactly the probability of A occurring anyways (=0.5/1), therefore knowledge/occurrence of B does not change the probability of A.

If you add a fifth event assigning now each outcome a probability of 0.2, then those proportions are not equal anymore.

We can generalize from your illustrative example and this geometric intuition to define independence as equal proportions, P(A|B) = P(A), that is, the probability of A happening (in general) equals the the probability of it happening (specifically) when B has happened, just as in your Venn diagram. Now P(A|B) = P(AB) / P(B), which is the proportion that A occupies in B. Substituting P(A) for P(A|B), we get the formal definition of independence for two events: P(AB) = P(A)P(B)