## Mutually Exclusive Events

Two events are mutually exclusive if they can’t happen at the same time.  For instance take flipping a coin – heads and tails are mutually exclusive events since you can only get one or the other.

When events are mutually exclusive, there is no overlap of their probabilities.  This means that if you want to work out the probability of either event A or event B happening, and they are mutually exclusive, then you can just add their individual probabilities:

Remember that the ‘U’ symbol is the union symbol, which is like ‘or’ – so in this case  means the probability of event A or event B happening.

### Expected number for a certain outcome

Sometimes you’re asked to estimate how many times you would expect to get a particular outcome, if you performed an experiment quite a few times.  A very simple example would be:

How many times would you expect to get ‘heads’ if you flipped a coin 100 times?

To work this out, you need to know the probability of the outcome happening for one experiment or trial, and then multiply this by the number of trials.

The probability of getting a ‘head’ from one coin flip is 50%, or 0.5.  What you do is take this ‘0.5’, and multiply it by the number of trials – 100 in this case:

There’s your answer – you’d expect to get 50 ‘heads’ if you flipped a coin 100 times.  Now of course, if you went out and flipped a coin 100 times, you wouldn’t necessarily get 50 heads.  You might get 43, or 54.  This is because probabilities are not certainties.  So even though we’d expect 50 heads, we could get more or less.

But in general, to estimate how many times you’d expect an event to happen, do this:

### Average run length of an outcome

Say you went and flipped a coin 10 times, and recorded what you got, in order:

H, T, H, H, T, H, T, T, H, H

If you look at the sequence you recorded, you can see that there are ‘runs’ of heads – sections in the sequence where only heads occur.  I’ve shown these runs here:

So reading from left to right, there’s a run of 1 head, then a run of 2 heads in a row, then another of 1, and finally a run of 2 heads in a row.  You can find the average run length by adding up these run lengths and dividing by how many different runs there are:

### Average run lengths for large numbers of trials

When you do a certain trial or experiment lots and lots of times, there is a formula you can use to calculate the average run length of a particular event:

You take 1, and divide it by 1 minus the probability of the event happening in one trial.  In the case of flipping a coin, the probability of getting a ‘head’ in one trial is 0.5.  So the average run length of heads if you flip a coin a lot of times is:

This surprises some people who expect that you would get a shorter average run length from flipping a coin.

This formula only works when the trial or experiment is repeated lots of times (I use 100 as being ‘lots’), and when the outcome of each trial is independent of any other trials.