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.