1 Introductory probability

Probability as an informal idea is something you will have been familiar with for a long time. In conversation with friends, you must have used sentences such as

• ‘It might start raining soon’
• ‘I might be lucky and pass all my examinations’
• ‘It is very unlikely that my team will not win the Premiership this year’
• ‘Getting a good degree will improve my chances of getting a good job’

Essentially, when you are talking about whether some event is likely to happen, you are using the concept of probability. In reality, we need to agree on some terminology so that misunderstanding may be avoided.

1.1 Terminology

To start with there are four terms $-$ experiment, outcome, event and sample space $-$ that need formal definition. There will, of course, be others as you progress through this Workbook.

1. Experiment: - an activity with an observable result, or set of results, for example
   1. tossing a coin, the result being a Head or a Tail
   2. testing a component, the result being a defective or non-defective component
   3. maximum speed testing of standard production cars;
   4. testing to destruction armour plating intended for use on tanks.

   Some of the experiments outlined above have a very limited set of results (tossing a coin) while others (destruction testing) may give a widely variable set of results. Also it is worth noting that destruction testing is not appropriate for all products. Companies manufacturing say trucks or explosives could not possibly test to destruction on a large scale - they would have little or nothing left to market!

2. Outcome - an outcome is simply an observable result of an experiment, for example
   1. tossing a coin, the possible outcomes are Heads or Tails
   2. testing a component, the outcome being a defective or non-defective component
   3. maximum speed testing of standard production cars, the outcomes being a set of numbers representing the maximum speeds of a set of vehicles
   4. testing to destruction armour plating intended for use on tanks, the outcomes might be (for example) the numbers of direct hits sustained before destruction.
3. Event - this is just an outcome or set of outcomes to an experiment of interest to the experimenter.
4. Sample Space - a sample space is the set of all possible outcomes of an experiment.

For example, if we throw a die then the sample space is $\left\{1,2,3,4,5,6\right\}$ and two possible events are

1. a score of 3 or more, represented by the set: $\left\{3,4,5,6\right\}$
2. a score which is even, represented by the set: $\left\{2,4,6\right\}$ .

   Everyday examples include games of chance.

Example 7

Obtain the sample space of the experiment throwing a single coin.

Solution

Consider the experiment of throwing a coin which can land Heads up ( $H$ ) or Tails up ( $T$ ). We list the outcomes as a set $\left\{H,T\right\}$ $-$ the order being unimportant. $\left\{H,T\right\}$ is the sample space. On any particular throw of a coin, Heads or Tails are equally likely to occur. We say that, for a fair coin, $H$ and $T$ are equally likely outcomes .

If the sample space can be written in the form of a list (possibly infinite) then it is called a discrete sample space (e.g. number of tosses of a fair coin before Heads occurs). If this is not possible then it is called a continuous sample space (e.g. positions where shells land in a tank battle).

Task!

List the equally likely outcomes to the experiments: 

1. throwing a fair die with six faces labelled 1 to 6

   [Note: ‘die’ is the singular of ‘dice’, although most people use ‘dice’ instead.]

2. throwing three fair coins.

Answer

Task!

For the following list of experiments, list (if possible) a suitable sample space. If you cannot write out a suitable sample space, describe one in words.

1. Test a light switch
2. Count the daily traffic accidents in Loughborough involving cyclists
3. Measure the tensile strength of small gauge steel wire
4. Test the maximum current carrying capacity of household mains cabling
5. Test the number of on-off switchings that a new type of fluorescent tube will cope with before failure
6. Pressure test an underwater TV camera.

Answer

Example 8

A car manufacturer offers certain options on its family cars. Customers may order:

1. either automatic gearboxes or manual gearboxes
2. either sunroof or air-conditioning
3. either steel wheels or allow wheels
4. either solid colour paint or metallic paint

Find the number of outcomes in the sample space of options that it is possible to order and represent them using a suitable diagram.

Solution

A suitable diagram is shown in Figure 6. The diagram makes it easy to find the number of outcomes simply by counting. It also points the way to a formula for calculating the number of outcomes.

Figure 6 :

In this case there is a total of 16 outcomes in the sample space of options. Note that in each case the customer makes two choices. This implies that there are

$2\times 2\times 2\times 2=16$

options in total.

Diagrams such as the one above are called tree diagrams . They are only suitable in simple situations.

1.2 Events

As we have already noted a collection of some or all of the outcomes of an experiment is called an event . So an event is a subset of the sample space. For example, if we throw a die then the sample space is $\left\{1,2,3,4,5,6\right\}$ and two possible events are

1. a score of 3 or more, represented by the set: $\left\{3,4,5,6\right\}$
2. a score which is even, represented by the set: $\left\{2,4,6\right\}$ .

Example 9

Two coins are thrown. List the ordered outcomes for the event when just one Tail is obtained.

Solution

$\left\{H,T\right\},\left\{T,H\right\}$

Note that here the order does matter unlike for sets in general.

Task!

Three coins are thrown. List the ordered outcomes which belong to each of the following events.

1. two Tails are obtained
   
   
2. at least two Tails are obtained
   
   
3. at most two Tails are obtained

   State the relationship between 1. and 2. and that between 1. and 3.

Answer

Task!

A new type of paint to be used in the manufacture of garden equipment is tested for impact shock resistance to damage and scratch resistance to damage. The results (50 samples) are as follows

If $A$ is the event $\left\{\text{High Shock Resistance}\right\}$ and $B$ is the event $\left\{\text{High Scratch Resistance}\right\}$ , describe the following events and determine the number of samples in each event.

1. $A\cup B$
2. $A\cap B$
3. $A^{\prime }$
4. $B^{\prime }$

Answer

1.3 Complement

We have met the complement before (Section 35.1 page 5) in relation to sets. We consider it again here in relation to sample spaces and events. The complement of an event is the set of outcomes which are not members of the event.

For example, the experiment of throwing a 6-faced die has sample space $S=\left\{1,2,3,4,5,6\right\}$ .

The event “score of 3 or more is obtained” is the set $\left\{3,4,5,6\right\}$ .

The complement of this event is $\left\{1,2\right\}$ which can be described in words as “score of 3 or more is

not obtained” or “score of 1 or 2 is obtained”.

The event: “even score is obtained” is the set $\left\{2,4,6\right\}.$

The complement of this event is $\left\{1,3,5\right\}$ or, in words “ even score is not obtained” or “odd score

is obtained”.

In the last but one Task concerning tossing three coins:

• the complement of event 1. is $\left\{TTT,THH,HTH,HHT,HHH\right\}$ ,
• the complement of event 2. is $\left\{THH,HTH,HHT,HHH\right\}$
• the complement of event 3. is $\left\{TTT\right\}$ .

Task!

State, in words, what are the complements of each of the following events in relation to the experiment of throwing three coins (avoid using the word not ):

1. two Heads are obtained   
2. at least two Heads are obtained   
3. at most two Heads are obtained.

Answer

Notation: It is customary to use a capital letter to denote an event. For example, $A$ = {two Heads are thrown}. The complementary event is denoted $A^{\prime }$ .

Hence, in the case where $A$ = {at least two Heads are thrown}, $A^{\prime }$ is the event {fewer than two Heads are thrown}.