# Find a counterexample for a false statment about independent events.

1. Oct 13, 2011

### spyrustheviru

"Construct a sample space to show that the truth of this statment P(A$\bigcap$B$\bigcap$C)=P(A)*P(B)*P(C) is not enough for the events A,B,C to be mutually independent.

Hint: Try finite sample spaces with equally likely simple events."

So, my though is that I need to find a sample space with 3 events A, B, C, that are not independent, yet P(A$\bigcap$B$\bigcap$C)=P(A)*P(B)*P(C) is true for them.
But I have already tried a couple of simple things and I can't seem to find a proper one. My problem propably lies in the events I take, not the spaces. I tried to use 2 tosses of fair, 6 sided dice, but my events were independent, 10 cards with the numbers 1-10 on them, 2 draws, and the card does not return to the deck. That one had dependent events, but the above statment was not true, and lastly, 2 coin tosses, but again, dependent events, untrue statment.

Any ideas?

2. Oct 13, 2011

### Citan Uzuki

Well, the easiest way to do this would be to let one of the events be the empty set -- then the equation P(A∩B∩C)=P(A)*P(B)*P(C) is automatically true.

Last edited: Oct 13, 2011
3. Oct 14, 2011

### mathman

Useful example. Whole space has numbers 1 through 8, each with probability = 1/8.
Let A = B = {1,2,3,4}
Let C = {1,5,6,7}
P(A)=P(B)=P(C)=1/2
P(A and B and C)=P({1})=1/8.

However A and B are obviously not independent. Also A and C are not independent.