Probability

We can talk about the chance or the risk that something will happen.

 

• What is the chance of winning the lottery?
• What is the risk of becoming ill?

 

In both cases one can also say: What is the probability that it will happen?

 

Probability theory is about calculating how likely it is that a particular event will occur.

It can be done in two ways:

 

• By means of statistics – either by using existing data or by carrying out new experiments and observations.
• By means of mathematics – for example when calculating the probability of rolling a six with a die: \(\tfrac{1}{6}\), because there are 6 sides \(\tfrac{6}{6}\), and one of them is the six.

 

Limits of probability

Probability is always expressed as a number between 0 and 1.

 

• \(P = 0\) means the event is impossible (for example rolling 8 with a die).
• \(P = 1\) means the event is certain (for example rolling less than 7 with a die).
• Other probabilities lie between 0 and 1 – for example \(P = 0{,}5\) corresponds to a 50 % chance.

 

Sample space

A sample space is the set of all possible outcomes. When rolling a die, the sample space is

 

\(\Omega = \{1, 2, 3, 4, 5, 6\}\)

 

Each number in the sample space is called an outcome.

For a fair die the sample space is uniform, because all outcomes have the same probability.

A non-uniform sample space could be the lottery, where the probability of winning is much smaller than the probability of losing.

 

Event

An event is what actually happens when you perform an experiment.

 

• Rolling \(2\) with a die is an event.
• Rolling an odd number is also an event: \(\{1,3,5\}\).

 

One also talks about:

 

• Impossible event: for example rolling \(8\) with a die.
• Certain event: for example rolling less than \(7\) with a die.