Probability is one of the hardest, most confusing topics in mathematics.  But if you learn the simple techniques and how to apply them, you can usually break down larger problems into several easy ones.

Relative frequency

Say I throw a die 20 times and it comes up with a six 5 of those 20 times.  The relative frequency for this situation would be 5/20 = 0.25. 

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In general, the relative frequency is:


This can also be expressed as:


Each time something happens, such as a die being tossed or a coin flipped, it is called a trial


Say I design an experiment to find out the probability of a coin coming up heads or tails.  The experiment could be as simple as tossing a coin 100 times and taking note of how many times heads came up, and the same for tails.

Each time I tossed a coin, which face turned up would be described as an outcome

Together all the possible outcomes are called the sample space.  For a coin the sample space is heads or tails.

Say I was tossing a die, and I was interested in when either a one or a two came up.  The 1 and the 2 would be called an event.  Whenever I tossed this die, the probability of either a 1 or 2 coming up would be 1/3.  The probability of this event would be 1/3.

Tree Diagrams

Tree diagrams are a graphical way of showing a die being thrown or a coin being tossed.  An example tree diagram for a coin being tossed 3 times is:

First of all, the diagram reads from left to right.  It starts at a point on the left, then splits into two branches – one representing (H)eads, the other (T)ails.  The number along the branch is the probability of that branch occurring.  In this case heads has the same probability as tails of occurring, which is expected.  The first split represents the first toss of the coin.

These two branches then each split into another two branches – this represents the second coin toss. The third coin toss is represented by the last column of branches – there are now 8.

You can use tree diagrams to answer probability questions.

Tree probability question

What is the probability of 3 heads occurring when I toss a coin 3 times?


Start off at the left hand side of the tree diagram.  We must find a path through the diagram that is all heads – there is only one. 

To work out the probability of this path occurring, you multiply the probabilities along the path by each other:


So there is a 12.5 % probability of all heads occurring when a coin is tossed 3 times.  This method is called the multiplication rule.

A different type of question might be:

Another tree probability question

What is the probability of rolling at least two tails?


For this type of question, you need to find all the paths through the diagram that have at least two tails.

There are 8 possible paths through the diagram, and of these 4 have at least two tails.  To work out the probability, there are two approaches:

1.  The easy one – since we know that each of the eight paths is equally likely, 4 out of 8 means a 50 % probability.

2.  Work out the probabilities for each of the four paths, and add them together:

                                            HTT – 0.5 ´ 0.5 ´ 0.5 = 0.125

                                            THT – 0.5 ´ 0.5 ´ 0.5 = 0.125

                                            TTH – 0.5 ´ 0.5 ´ 0.5 = 0.125

                                            TTT – 0.5 ´ 0.5 ´ 0.5 = 0.125