Ben Challenor

Posted on Monday, 29 March, 2004 - 04:58 pm:

There are 28 dominoes, with between 0 and 6 spots on each end. Turning a domino around does not make it a different domino.

Event A occurs when the spots on a domino add up to 6. It is true for 4 dominoes.
Event B occurs when the domino is a double. It is true for 7 dominoes.

They are not mutually exclusive. Event A and Event B occur together on one domino (3,3).

Now, are Events A and B independent?

Initially I thought not, because of the above. But if event A is given, you have 4 dominos to choose from, and one of them is a double. Therefore the probability of B is still 1/4.

So, even though this would not be the case with most examples, can you say that A and B are independent?

Cheers
Ben

Tristan Marshall

Posted on Monday, 29 March, 2004 - 06:22 pm:

Yes.

Independence is actually a slightly strange property in many ways. Our intuitive notion that $A$ and $B$ are independent if '' $A$ and $B$ don't affect each other'' doesn't cover all the possibilities. We can fix things so that '' $A$ and $B$ do affect each other'', but we still have $P (A \cap B) = P (A) P (B)$ , i.e. $A$ and $B$ are independent (this is what's happening in the above example).

The only property that matters when determining independence of $A$ and $B$ is whether $P (A \cap B) = P (A) P (B)$ , where $P (A \cap B)$ is the probability of $A$ and $B$ happening.

Ben Challenor

Posted on Monday, 29 March, 2004 - 06:40 pm:

Thanks for clearing that up!