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I have few doubts regarding the independent events.

  1. I am aware that we can put mutually exclusive events on a Venn diagram showing that two events do not overlap. Can we draw Venn diagram for independent events? What it will look like?
  2. I am aware that
    • P(E or F) = P(E) + P(F) - P(E and F)
    • P(E or F) = P(E) + P(F) --- if E,F are mutually exclusive
    • P(E and F) = P(E).P(F) --- if E,F are independent
    • But what about P(E or F) --- when E,F are independent?
  3. Can independent events be mutually exclusive also and vice-versa? Please provide reason and example.
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For question $1$, you can draw a Venn diagram for independent events, however you will not be able to tell if the events are independent by looking at the Venn diagram, as it will just look like a standard Venn diagram. (standard meaning how a Venn diagram would look for $2$ events $A,B$, that are not mutually exclusive)

For question $2$, if $A$, $B$ are $2$ independent events then $P(A\cup B)=P(A)+P(B)-P(A)P(B)$

For question $3$, a mutually exclusive events are necessarily dependent events (assuming the probability of both events is greater than $0$).

Proof:

Recall the following:

Let two events, $X$ and $Y$ be independent. Then it follows that $P(X \cap Y)=P(X)P(Y).$ These events are mutually exclusive if $P(X \cap Y)=0.$ Lastly remember that $P(X)>0$ and that also $P(Y)>0$, as we are discussing probability and it ranges from $0-1$.

So, since we know that $P(X)>0,P(Y)>0$, then it follows that $P(X)P(Y)>0.$ If these events were independent then $P(X)P(Y)=P(X\cap Y)>0$, but this would mean that they aren't mutually exclusive.

Therefore, the events can not be independent and mutually exclusive simultaneously if both their probabilities are more than $0$.

$Q.E.D.$

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  • $\begingroup$ Sujaan, could you please give a real life example? It will help in understanding it more clearly. $\endgroup$
    – sam
    Aug 1, 2013 at 19:01
  • $\begingroup$ Take flipping a coin and rolling a die for a pair of independent events. The die roll doesn't impact the coin and the coin doesn't affect the die and so this would be a relatively simple example to set up and run. $\endgroup$
    – JB King
    Aug 1, 2013 at 20:56

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