I have always loved maths from when I first started school. I am not sure why. All I know is that I have always loved working with numbers. I know that this would be many people's idea of a living nightmare but don't worry I don't intend to lecture anybody on why maths is the most amazing and brillliant subject of all! (I'll have to wait until I have enough time to write the reasons down first :o) ). However I think that there are too few sites around that focus on the less formal and more practical side of the subject and show that maths can be a lot of fun.

My favourite area is probability and statistics and so I felt it was appropriate that a little corner of my site should be devoted to this. If you are a student or you are looking for information then you should find some useful links in my study tips section but this page just contains a few basic facts that those who have little or no experience in maths may find surprising.

Believe it or not, the above answer is absolutely true. Let's see why that is.

If we assume that there are only 365 possible birthdays (forget about leap years) then it is not that difficult a task to work out the probability that everyone in the room of 40 has a different birthday.

Firstly though we need to know a basic probability rule. Imagine that we want to arrange 3 coloured frames (red, green and blue) on a wall. We have 3 choices when we hang the first one up (we could hang the red one up or the green one or the blue one). Then when the first one is already up on the wall we have 2 remaining frames to choose from and when we only have one frame left to hang up on the wall we only have one choice since there is only one left. There is a special rule that says that if you want to find the number of ways of doing a number of things, you multiply the number of ways you can do the first thing by the number of ways you can do the second thing etc. The total number of ways to arrange the frames is given by 3x2x1 (using our special rule).

Now, If we apply this logic to the 40 people in the room we can work out how many possibilities there are for everyone's birthday. The first person could have any one of 365 days as their birthday. Another way of saying this is that they are 365 choices for the first person's birthday. Now let's deal with the second person. He could have the same birthday as the first person so there are also 365 days that could be the second person's birthday. Continuing in this way we see that there are 365 choices for everyone's birthday. So, the total number of choices for all 40 people's birthdays is 365x365x365x......365 where 365 is multiplied 40 times. We can write this as 365^40 (365 to the power of 40).

Now let's assume that everyone in the room has a different birthday. There are still 365 choices for the first person's birthday, but when we move along to the second person one birthday (the first person's birthday) is gone, so there are only 364 choices left. Continuing in this way we see that the number of choices for the 40 people if everyone has a different birthday is 365x364x363x362x361x...x326 (keep subtracting 1 from 365 until you have 40 numbers written down and multiply them all together).

Now let's go back to our 3 coloured frames for a moment. say we want the red one to be first up on the wall. How many ways could this happen? Well, if the red one is first then the other two frames are second and third but the order could be red, green and blue or red, blue and green. So there are 2 ways that this could happen. We already know that there are 6 ways of arranging the frames in total. So we could say that the chance or probability of the red frame coming first is 2/6 or 1/3 or 33%.

We can do the same thing with the 40 people in the room. We know that there are 365x364x......x326 ways that the 40 people could all have different birthdays. We also know that the total no of choices for 40 people's birthdays is 365^40. So continuing as we did in our frames example the chance or probability of all 40 people having different birthdays is [ 365x364x...x326 ]/[ 365^40 ]. If you work this out on a computer (don't use your calculator-it will probably explode) you will get an answer of about 11% (11/100).

Right, we are nearly there. I hope you are still with me but don't worry if you are not. If you are not used to thinking this way probability can cause a mental breakdown! To finish off, think about what the weather will be like tomorrow. Will it rain? If only we knew! Yet there is one thing we can be certain of - it will either rain or it won't. Think about that for a second. It seems obvious but it means that if we know that the probability that it will rain is 70% we can quickly work out what the probability of no rain is. It will either rain or it won't so the 2 probabilities added together must equal 100%. So the probability that it won't rain is equal to 100% - 70% = 30%.

This means that we can work out the answer to the birthday problem. If there are 40 people in a room they will either all have different birthdays or at least 2 will have the same birthday. The 2 probabilities must sum to 100%. We worked out that the probability that they all have different birthdays is about 11%. So that means that the probability that at least 2 of the 40 have the same birthday is 89%!

So there you have it - strange but true! Unfortunately I can't tell you the probability of being able to win money off your friends who don't believe it is true by getting them to put their money where there mouth is! But give it a go and you just might become a millionaire ;o).