Probability - Introduction - SS2 Mathematics Lesson Note

Probability is that branch of mathematics which deals with the possibilities or chances of events occurring. Such as the chance that rain will fall today or the chance that your favourite football will win their next game. Probability values exist on a range between \(0\) (meaning the event has no chance of happening) and \(1\) (the event will occur with certainty). Usually, most events have probabilities in a fraction or decimal form, that is, \(\frac{1}{6}\) or \(0.17\). The lower the probability value, the less likely that event is to occur and vice versa.

Mathematically, probability is the measure of an event taking place under certain laws of chance. Every possibility is an outcome. Usually, each outcome has an equal chance of happening, meaning they are equiprobable. For example, a die with \(6\) faces has an equiprobable chance of landing any of the \(6\) faces.

The set of all possible outcomes is called the equiprobable sample space. Events whose occurrence excludes all other outcomes at the same time is known as a mutually exclusive outcome. When a coin is tossed for instance, obtaining a head means you can’t obtain a tail at the same time. An independent outcome means two or more events have no influence over each other, such as taking a red ball out of bag of balls and putting it back before picking randomly from the same bag.

The probability of an event is equal to the number of possible outcomes over total outcomes. The probability of an event \(A\), \(P(A)\) is

\(P(A) = \frac{number\ of\ equiprobable\ favourable\ outcomes}{number\ of\ all\ equiprobable\ outcomes}\)

Note, the probability that an event not occurring is equal to the probability of the event occurring subtracted from \(1\), that is \(P\left( A^{'} \right) = 1 - P(A)\). \(A'\) is read as “prime \(A\)”.

Example: A die is tossed once, what is the probability that an even number is obtained as the result?

Solution

\[equiprobable\ smaple\ space = \{ 1,2,3,4,5,6\}\]

\[favourable\ equiprobable\ outcomes = \{ 2,4,6\}\]

\[P(even\ number\ face) = \frac{number\ of\ equiprobable\ favourable\ outcomes}{number\ of\ all\ equiprobable\ outcomes} = \ \frac{3}{6} = \ \frac{1}{2}\]