Are mutually exclusive events independent? - YouTube

Let's investigate the question: Are mutually exclusive events independent?

First note that independence and mutual exclusivity are different concepts; they mean very different things.

But some students confuse them, and working through this question might help to resolve some doubts.

In this video I assume that you've been introduced to the concepts of conditional probability and independence,

but I will do a very brief review of those concepts.

Here's our formal definition of independence:

Events A and B are independent if and only if the probability of their intersection is equal to the product of the individual probabilities.

You might have seen independence defined in terms of conditional probabilities, which leads to a more intuitive explanation,

and is a reasonable approach, but mathematically we run into a bit of a snag.

Here's the conditional probability formula for event A given event B,

but this is not defined if the probability of B is 0.

Similarly, here's the conditional probability formula for event B, given event A has occurred.

But this is not defined if the probability of A is 0.

So we run into snags in the 0 probability case,

and that 0 probability case is meaningful for the problem we're looking at in this video.

For the case where both probabilities are nonzero, these three statements are equivalent,

and they all mean that A and B are independent events.

Each of these three statements implies the others.

If one of these statements is true, then they are all true and A and B are independent.

If one is false, they are all false and A and B are not independent.

Now to the problem.

Here is a visual representation of events A and B.

Here, visually at least, A and B share part of the sample space.

And here is a visual representation of mutually exclusive events.

Here, A and B share no common ground in the sample space. They share no sample points and they cannot occur together.

Some students look at this and think that A and B are separate, that they're doing their own thing, and so they're independent.

But that's not what independence means in a probability sense.

Let's look at what this situation does tell us.

Here, the events are mutually exclusive so the probability of their intersection is 0.

But if the probability of the intersection is 0, that can only equal the product of the individual probabilities

if the probability of A is 0, or the probability of B is 0 (or both are 0).

So if A and B are mutually exclusive events,

they are independent if and only if the probability of A is 0 or the probability of B is 0.

But I'm not particularly interested in this special case of 0 probabilities in this situation,

and I think that the case where A and B both have positive probabilities of occurring is a little more meaningful.

So at the risk of being a little redundant, I'm going to take a closer look at that scenario.

Suppose that A and B are mutually exclusive events,

where the probability of A is greater than 0, as is the probability of B.

This of course implies that the product of their probabilities is greater than 0.

They are assumed to be mutually exclusive here,

so the probability of their intersection is 0,

and thus the probability of their intersection is not equal to the product of their individual probabilities.

This means that if A and B are mutually exclusive events with positive probabilities of occurring,

then A and B are not independent.

We could also phrase this in terms of conditional probability,

which might make a little more intuitive sense for some.

Here, if B happens what is the probability of A?

Well if we're in circle B, we cannot be in circle A,

so the probability of A given B is 0.

You can verify that with the conditional probability formula if you wish.

The original probability of A is greater than 0 here,

so the conditional probability of A given B is not equal to the probability of A,

and thus A and B are not independent.

The knowledge that B has occurred has changed the probability of A from whatever it was originally to 0,

A and B are very much dependent.

And of course we could easily switch B and A around, and make the same argument with the conditional probability of B given A.

If we are in circle A then we cannot be in circle B,

and thus the probability of B given A is 0.

We could have used any one of these three statements to draw this conclusion.