6.01 Describing chance and sample spaces

What is probability?

In day to day life, there are many places where the language of probability is used. For example:

• The chance of rain today is $75%$75%
• A medication has an effectiveness of $90%$90%
• $4$4 in $5$5 people agree that a particular brand of toothpaste is more effective than another
• I will probably go to the gym today

Probability the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the number of cases possible. That is, the probability of an event occurring is:

$P(event)=\frac{\text{number of favourable outcomes}}{\text{total possible outcomes}}$P(event)=number of favourable outcomestotal possible outcomes​

The range of values that all probabilities can take is between $0$0 and $1$1, where a probability of zero indicates the event cannot possibly occur and a probability of one indicates the event is certain to occur. This can be visualised as follows: 

 

 

If all the outcomes can be easily listed, then the process of counting favourable and total outcomes is relatively straightforward. When calculating probabilities for a large or complex set of outcomes we may wish to employ approaches such as tree diagrams or use properties of probability that relate to different characteristics of the event such as the complement of the event.

 

Sample Space

The sample space, sometimes called an event space, is a listing of all the possible outcomes that could arise from an experiment.

For example:

• Tossing a coin would have a sample space of $\left\{\text{Head},\text{Tail}\right\}${Head,Tail}, or $\left\{H,T\right\}${H,T}
• Rolling a dice would have a sample space of $\left\{1,2,3,4,5,6\right\}${1,2,3,4,5,6}
• Watching the weather could have a sample space of $\left\{\text{sunny},\text{cloudy},\text{rainy}\right\}${sunny,cloudy,rainy} or $\left\{\text{hot},\text{cold}\right\}${hot,cold}
• Asking questions in a survey of favourite seasons could have a sample space of $\left\{\text{Summer},\text{Autumn},\text{Winter},\text{Spring}\right\}${Summer,Autumn,Winter,Spring}

Notice how the the sample space is listed as a set using curly braces $\left\{\ \right\}${ }. We could also display all outcomes in a sample space by using a table (array) or tree diagram.

 

Event

An event is the word used to describe a single result from within the sample space. It helps to identify which of the sample space outcomes we might be interested in.

For example, these are all events.

• Getting a tail when a coin is tossed.
• Rolling more than $3$3 when a die is rolled
• Getting an Ace when a card is pulled from a deck

We use the notation $P\left(\text{event}\right)$P(event) to describe the probability of a particular event.

 

Practice questions

QUESTION 1

QUESTION 2

Question 3