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.

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 $0$).


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 $0$.


  • $\begingroup$ Sujaan, could you please give a real life example? It will help in understanding it more clearly. $\endgroup$ – Sumit Aug 1 '13 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 '13 at 20:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.