One of the topics that you will study in unit ½ Methods is probability. Probability is a bit different to the other topics you will have covered in maths, as doing well relies a lot on your comprehension of the question. Despite this, it is a topic that is easy to do well in once you get the hang of it.
This article will give you an overview of everything you need to know about probability, so keep reading!
Basic Terminology
As a refresher, lets cover some basic terminology. The sample space for an experiment lists the set of all the possible outcomes that can arise. An event is a subset of the sample space – this is a specific outcome that can arise from the experiment.
For example, is standard dice is rolled, the sample space would be S={1,2,3,4,5,6}. An event may be rolling an even number or getting the number five.
The probability of an event occurring is defined as:
Pr(A)=Number of Favourable Outcomes/Total Number of Outcomes
Also, the union between two sets is denoted by ∪, which represents all the elements in multiple sets combined. The intersection between two sets is denoted by ∩ , which is the elements that two sets may have in common.
A compound event is when multiple events occur simultaneously or in succession. There are multiple outcomes that can arise from that event occurring.
And finally, the complement of an event is it exact opposite. They occur when there are only two outcomes, like passing a test and failing a test.
An event and its complement make up the entire sample space, so:
Pr(A')=1-Pr(A)
The Addition Rule for Probabilities
The addition rule is simply used to find the probability of either one of two events occurring. Firstly, the probabilities of each event occurring is added together. Then, the probability of both events occurring at the same time is subtracted to prevent double counting.
Pr(A∪B)=Pr(A)+Pr(B)-Pr(A∩B)
Once you know three of these terms, you can rearrange the expression to find the other.
From this, we can find the expression for mutually exclusive events. These are events that cannot occur at the same time, meaning that have no intersection.
Therefore if:
Pr(A∩B)=0
Thus, the addition rule simplifies to:
Pr(A∪B)=Pr(A)+Pr(B)
Conditional Probability
Conditional probability gives the chance of an event occurring, given that another event has already occurred. For example, we might want to calculate the probability of someone taking their dog for a walk, if it rains on that day.
The probability of event ‘A’ occurring, given that ‘B’ has already occurred is given as follows:
Pr(A|B)=Pr(A∩B)/Pr(B)
It is important that you keep the order of terms correct when using this formula, and make sure you read the question careful to work out which event has already occurred. Students get confused with this all the time!
The probability of event ‘B’ occurring, given that ‘A’ has occurred is given as follows:
Pr(B|A)=Pr(A∩B)/Pr(A)
As you can see, the formula looks slightly different.
Independent and Dependent Events
Two events are independent if the occurrence of one event does not affect the chances of the occurrence of the other event. This is commonly associated with events with replacement.
Let’s say we had six marbles in a bag – two were pink, one was purple and three were blue. Randomly, two marbles were chosen out of the bag and the colour was recorded – each one was replaced in the bag after it was chosen. This is an example of an independent event, as the colour taken out first will not affect the colour taken out second.
On the other hand, let’s say we chose a purple marble, and then didn’t put it back, and then chose a pink marble. Since there was only one purple marble in the bag, it is impossible to choose a purple marble again on the second go. This is an example of a dependent event, as choosing a purple marble on the first go affected the chances of choosing a purple marble on the second go. As such, dependent events are done without replacement.
The following rule holds for independent events:
Pr(A∩B)=Pr(A) x Pr(B)
They also have this property:
Pr(A|B)=Pr(B)
Representing Data
There are multiple ways of representing the outcomes of an experiment, and the three most used ones are Venn Diagrams, Tree Diagrams and Two Way Tables.
Tree diagrams are used to visualise all the outcomes that can arise from a multistep experiment.
If we had the letters from the word BAT on four cards, and we drew two cards out randomly, the tree diagram may look like either of the following:
The first diagram shows the outcomes if the cards were replaced after each trial, while the second one shows the outcomes if the cards weren’t replaced. The sample space can be written by following along each branch.
Venn diagrams and two way tables illustrate the relationship between two or more data sets.
Venn Diagrams look like this:
The big circle on the left shows the number of elements that are only in set A, and the big circle on the right shows the number of elements that are only in set B. In the middle, you see how many elements are in both sets, where they overlap. The number on the outside of the circles represents the elements that belong in neither set.
And here’s a handy template you can use when filling out a two way table.
Just remember that all the rows and columns must add up.
Tips and Tricks
- Be mindful about the way the question is worded. Make sure you get a sound understanding of the situation before doing any calculations.
- If you need to, define any variables appropriately.
- Leave all answers as exact values, unless otherwise specified.
- Practise converting between fractions, decimals and percentages as probabilities can be expressed in either one of these ways.
- If you get a probability that isn’t between 0 and 1, you’ve likely done something wrong.
- Become familiar with the different notations and symbols that are used.
Hopefully this article taught you a thing or two about probability! As difficult as it might seem at first, it is a topic that you can certainly do well in.
If you’d like to download a set of summary notes covering this topic, click here.
