Probability - Independence Question

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SUMMARY

This discussion centers on the concept of independent events in probability, specifically the graphical representation using Venn diagrams. The key mathematical principle established is that for two independent events A and B, the probability of their intersection is given by P(A ∩ B) = P(A)P(B). The confusion arises when trying to visualize this relationship in a Venn diagram, as it resembles diagrams for dependent events. Additionally, the distinction between independent and mutually exclusive events is clarified, emphasizing that mutually exclusive events cannot occur simultaneously, thus making them dependent.

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  • Understanding of basic probability concepts
  • Familiarity with Venn diagrams
  • Knowledge of independent and dependent events
  • Mathematical notation for probability (e.g., P(A), P(B))
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  • Study graphical representations of independent events using Venn diagrams
  • Explore the mathematical properties of independent events in probability theory
  • Learn about mutually exclusive events and their differences from independent events
  • Practice problems involving the calculation of probabilities for independent events
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Students and educators in mathematics, statisticians, and anyone seeking to deepen their understanding of probability theory and the nuances between independent and dependent events.

jmcgraw
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I'm trying to get the idea of independent events well grounded in my mind, but I'm having some difficulty.

First of all, what would a venn diagram for independent events look like? I know you have the two events intersecting, and the intersection is equal to the product of the probabilities of the two events... But all I can picture is a normal diagram of two events intersecting, which by eyeballing looks just like the diagram of any 2 events (depndent or independent). Is there something unique to the diagram of independent events?

Mathematically, I just need to know that [tex]P(A \cap B) = P(A)P(B)[/tex] . But I can't form a graphical intuition about this, which I think would help me a lot.

Incidentally, I can't help but keep picturing the venn diagram of mutually exclusive events as a representation of independent events... But I know that is dead wrong, since mutually exclusive events are dependent.

Can anyone help me clarify this in my mind? Thanks!
 
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If B is independent of A, it means that the intersection of B with A is the SAME fraction of A's total area as the intersection of B with not-A is of not-A's area.
 
If B is independent of A, it means that the intersection of B with A is the SAME fraction of A's total area as the intersection of B with not-A is of not-A's area.

I think I just got more confused. :smile:

Isn't the intersection of B with not-A the same as not-A? giving the fraction to be simply 1?

Obviously I must be missing something, big time.
 

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