10.03 Theoretical probability

Probability terms and notation

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

The notation $P\left(A\right)$P(A) means 'the probability that event $A$A occurs. All probabilities lie on the interval $0\le P\left(A\right)\le1$0≤P(A)≤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: 

 

Before we calculate probabilities, let's familiarise ourselves with the language and notation for describing components in this topic. In probability, the sample space is a list of all the possible outcomes of an experiment.

Outcomes are the results of an experiment or trial. For example, think about flipping a coin. There are two possible outcomes - a head or a tail. So when we list (or write out) the sample space, we write:  

$S=\left\{heads,tails\right\}$S={heads,tails}

We can write a sample space using a list, table, set notation as above or a diagram such as a Venn or Tree diagram.

An event is a subset of the of the sample space and is often represented by a capital letter to abbreviate the description of the event in calculations.

For example, we could have the experiment of rolling a six sided dice and let $E$E = the event of rolling an even number, $O$O = the event of rolling an odd number and $A$A = the event of rolling a number less than $3$3.

Then our sample space is:

$S$S

$=$=

$\left\{1,2,3,4,5,6\right\}${1,2,3,4,5,6}

And the events can be written as the sets:

$E$E	$=$=	$\left\{2,4,6\right\}${2,4,6}
$O$O	$=$=	$\left\{1,3,5\right\}${1,3,5}
$A$A	$=$=	$\left\{1,2\right\}${1,2}

 

If all the outcomes can be easily listed, then the process of counting favourable and total outcomes is relatively straightforward.

For example, say we wanted to know the probability of rolling a $3$3 on a die. There are six possible outcomes in total (the sample space is $\left(1,2,3,4,5,6\right)$(1,2,3,4,5,6)). However, there is only one three on a die. So, we write our answer as $P(3)=\frac{1}{6}$P(3)=16​

There is an assumption in the work we are doing that every outcome is equally likely to occur. This allows us to utilise the formula above for the probability of an event occurring. This is also known as theoretical probability, which is the expected probability based on knowledge of the system and determining the number of favourable outcomes and number of total possible outcomes mathematically. 

Experimental probability with large number of trials

The graph below shows the results of tossing a coin $50$50 times. As we can see the probability varies significantly at the start but as we increase the number of coin tosses the probability settles and approaches the theoretical probability of $0.5$0.5.

Practice questions

Question 1

question 2

Complementary events

A complement of an event are all outcomes that are NOT the event. If $A$A is the event then the complement is denoted as $\overline{A}$A or $A'$A′ .

The following are examples of events and their complements:

• If event $A$A is tossing a coin and getting $\left\{Heads\right\}${Heads}, the complement $\overline{A}$A is $\left\{\text{not a head}\right\}${not a head} which is $\left\{Tails\right\}${Tails}
• If event $B$B is rolling a $6$6 sided die and getting $\left\{2\right\}${2}, then the complement $\overline{B}$B is $\left\{1,3,4,5,6\right\}${1,3,4,5,6}

Complementary events have the property that their probabilities always add to $1$1. Since it is certain either event $A$A occurred or it did not occur. That is:

$P\left(A\right)+P\left(\overline{A}\right)=1$P(A)+P(A)=1

In some cases it may be easier to count the possibilities where an event does not happen and use this fact to find the probability that the event does happen more efficiently. 

Practice questions

Question 3

Question 4