13A - Introduction to probability

Learning intentions:

In this section we will examine:

Different types of common sample spaces.
Random experiments and events that occur within the sample space.
The classical definition for computing probabilities.
Complementary events of a random experiment.

Random experiments and their sample spaces

A random experiment is a replicable situation in which the outcome is considered 'random' as we have no way of knowing what the outcome will be until the experiment is run. A trial is a single run of the experiment which results in a observable outcome. The observable outcome will be one of a set of know possible outcomes for the ransom experiment. The set of all possible outcomes is known as the sample space in probability.

Consider flipping a coin:

The random experiment is flipping a coin.
A trial would be an individual coin toss.
The coin toss may result in a heads (H) or tails (T) - this is the sample space {H,T}.

The random experiment involving a coin is, perhaps, the most simple example. A variety of examples are listed in the table below:

Examples of random experiments and their sample spaces

The Greek letter epsilon,

$\varepsilon \!$

, is used to represent the sample space.

Random Experiment:

The upper-facing number when a die is rolled.

The number of seats occupied in a 56-seat bus.

The outcome of flipping two coins.

The number of coin tosses until the first head.

The life time of a light-bulb.

Sample Space:

For revision of set notation please refer to section 1A.

Discrete or continuous

A sample space (and events) may be discrete or continuous.

A discrete sample space, or event, can only take on a well defined set of outcomes. For example, the number of seats occupied on a bus is a discrete set - it is not possible to have 32.87 seats filled.
A continuous sample space, or event, can take on any value within a given range. For example, the lifetime of a light bulb can take on any value the only limiting factor is the instruments used to measure the time - it is possible that the bulb would last 19.19287 minutes.

Finite or infinite

A sample space (and events) may be finite or infinite.

A finite sample space, or event, can only take outcome within a set range. For example, rolling two dice only have four possible outcome - it is not possible to have any more outcome than this.
An infinite sample space, or event, can take on any value with at least one end unbounded. For example, the lifetime of a light bulb can take on any value from zero until infinity - here it is bounded at zero and unbounded at infinity.

Events

An event is a subset of outcomes within the sample space. An event may consist one outcome or it may consist of several outcomes. We commonly use capital letters (A,B,C,...) or plain language ('getting an odd number') to represent events.

Examples of events

Random Experiment:

The upper-facing number when a die is rolled.

Events(s):

Rolling a 5 or more:

The number of seats occupied in a 56-seat bus.

Having 12 or less passengers:

The outcome of flipping two coins.

Getting at least 1 tail:

The number of coin tosses until the first head.

Getting the first H in less than 5 rolls:

The life time, in minutes, of a light-bulb.

The light-bulb last longer than 2 hours

Probabilities

It is often possible to build simple models to describe the sample space (we will look at this in more detail in 13B). For this model it is then possible to assign probabilities to events. Before we go too much further we must look at two important properties of probability:

Property 1:

Property 2:

All probabilities, for events, must be between 0 and 1 (inclusive):

The sum of all the probabilities within the sample space always adds to 1:

Please note:

A probability of zero is possible and it indicates that there is no chance of that event occurring. Probabilities close to zero indicate only a small chance of occurring.
A probability of one is possible and it indicates that the event occurring is a certainty. Probabilities close to one indicate only a large chance of occurring.

Calculating probabilities using number of outcomes

When the number of outcomes in the sample space (

) is known, we can compute the probability of an event, A, using the following formula:

$\varepsilon \!$

Worked Example:

If we consider a fair six-sided die, we know there are six outcomes {1,2,3,4,5,6}. Therefore, each of the outcomes is equally likely:

Furthermore, If event A is "rolling less than a 3" (A = {1,2}), the associated probability is:

Calculating probabilities using area

The probability of an event can also be determined using areas. Similarly to using the number of outcomes, we can also determine the probability using the ratio of the desired outcomes area to the total area making up the sample space. Therefore:

Complementary events

When the union of two mutually exclusive events makes up the entire sample space the two events are said to be complementary. That is, if two events have no elements in common but when combined include all possible outcomes in the sample space they are complementary events. The complement of is A,denoted A' (sometimes A^c), is the set of all outcomes not contained within set A.

Figure 1 - The complement of A (A').

As A and A' are mutually exclusive there intersection is the null set (∅). Therefore:

Hence, the probability of A and A' will add up to 1 (together they are the sample space):

13A - Exercises:

Success criteria:

You will be successful if you can:

Identify the sample space (all possible outcomes for trials) of simple random experiments.
Calculate probabilities based off the number of desired outcomes.
Use the definition of complementary events to efficiently compute probabilities.