Maths Times Infinity > Topics > Probability
Probability
A very useful skill, calculating the chance of something occuring. An interesting area of mathematics indeed.
Probability is concerned with the *chance* that something will/won't happen. Combinatorics is concerned with the number of different/distinct ways in which something could happen. They are related because often the best way to calculate a probability is to add up how many ways something could happen and divide it by how many total different things could happen.
For example, a couple has two children. What's the probability that they have two boys? Well, there are 4 possibilities in total: BB, BG, GB, GG. Only one of them is the one we are concerned with. So there's a 1 in 4 chance. A common mistake is to not distinguish between the cases BG and GB.
Factorial
This concept comes up a fair bit in harder probability. It involves multiplying a whole bunch of numbers together...
#n!=1 \times 2 \times 3 \times \dots \times n#
The factorial of a number is the product of all the numbers from 1 up to itself.
The factorial function occurs quite often in probability. For example, if I hold a race between 8 contestants, the number of different rankings they could end up in is 8!. That's because 8 people could come first which leaves 7 who could come second, then 6 who might come third and so on until the only person left has to come last. So, #8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=8!=40,320#. If I told you that either Elliot or Gertie had won the race, there would only be #2 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=2 \times 7!=10,080# different possible outcomes.
Permutations
These combinations of different orderings are called permutations. Let's look at some trickier ones.
Now we have another race between 8 people but I only care who comes first, second and third. In how many ways can that happen?
Well there are #8 \times 7 \times 6=336# different results (ignoring 4th to 8th place). Another way of writing #8 \times 7 \times 6# is 8!/5! because #8!/5!=(8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)/(5 \times 4 \times 3 \times 2 \times 1)# and everything except the #8 \times 7 \times 6# part will cancel. So, in general you get the following formula.
nPr=n!/(n-r)!
Choose
Let's say 20 people audition for a role and I have to pick my three favourites (in no particular order). How many different groups of three could I choose?
Well, let's say I pick them in a specific order first. I have 20 choices for the first one, then 19, then 18. So that's 20*19*18 total ways.