Sample space and events

A sample space is the set of all possible outcomes in an experiment.

 

• Coin toss: \(\Omega = \{\text{heads, tails}\}\)
• Die: \(\Omega = \{1,2,3,4,5,6\}\)

 

Events

An event is one or more outcomes from the sample space.

 

• Rolling a 4 with a die: \(\{4\}\)
• Rolling an even number: \(\{2,4,6\}\)

 

We also talk about special events:

 

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

 

Uniform sample space

We speak of a uniform sample space when all outcomes are equally likely.

This could for example be a fair die.

 

Here the sample space is \(\{1, 2, 3, 4, 5, 6\}\), and all outcomes have the same probability.

The probability of rolling a six is:

 

\(\large \tfrac{1}{6}\) or one out of six

 

If the event is rolling an odd number, the favorable outcomes are \(\{1,3,5\}\).

The probability is:

 

$$ \large\tfrac{3}{6} \Leftrightarrow \tfrac{1}{2} \Leftrightarrow 0.5 \Leftrightarrow 50\% $$

 

Non-uniform sample space

In some experiments the probabilities are not equal. This is called a non-uniform sample space.

An example is the lottery, where the probability of winning is much smaller than the probability of losing.

 

Two dice

The probability of rolling a six with one die is \(\tfrac{1}{6}\).

But what if you have two dice and want the probability of at least one six?

 

There are in total \(6 \times 6 = 36\) possible outcomes.

The favorable outcomes can be shown in a table, where the first die is the horizontal axis and the second the vertical:

 

Favorable outcomes

	1	2	3	4	5	6
1						X
2						X
3						X
4						X
5						X
6	X	X	X	X	X	X

 

As you can see, crosses are placed in all outcomes with at least one six. There are 11 in total.

It may be tempting to think there are 12 favorable outcomes (6 for each die), but the outcome where both are sixes counts only once. This is important to remember!

 

The probability is therefore:

 

$$ \large \tfrac{11}{36} \approx 0.30 = 30\% $$

 

If the event instead was rolling two sixes, there is only 1 favorable outcome:

 

$$ \large \tfrac{1}{36} \approx 0.0277 = 2.77\% $$

 

Summary

• Sample space = all possible outcomes.
• Event = one or more outcomes.
• Probability = fraction of favorable outcomes compared to all possible outcomes.
• With more dice or experiments, the sample space grows quickly.