Probability Objectives

When you have competed it you should* know what a ‘sample space’ is

* know the difference between an ‘outcome’ and an ‘event’.

* know about different ways of estimating probabilities.

Key terms: Sample space, Event, Complement of event, Trial/Experiment, Outcome.

Probability is a measure of the likelihood that something happening.

Estimating Probability

There are three different ways of estimating probabilities.

Method A: Theoretical estimation: Use symmetry i.e. counts equally likely outcomes.e.g. The probability of head ( P(H) ) when a coin is tossed.

Probability

Estimating Probability

Method B: Experimental estimation: Collect data from an experiment or survey.e.g. What is the estimated probability of a drawing pin landing point upwards when dropped onto a hard surface.Method C: Make a subjective estimateWhen we cannot estimate a probability using experimental methods or equally likely outcomes, we may need to use a subjective method.e.g. What is the estimated probability of my plane crashing as it lands at a certain airport?

Sample space The list of all the possible outcomes is called the sample space s of the experiment

It is important in probability to distinguish experiments from the outcomes which they may generate.

Experiment Possible outcomes

Tossing a coin (H, T)

Throwing a die (1, 2, 3, 4, 5, 6)

Guessing the answer to a four multiple choice question

(A, B, C, D)

The complement of an event

The event ‘not A’ is called the complement of the event. The symbol A1 is used to denote the complement of A.

P(A) + P(A1) = 1

An Event is a defined situation.e.g. Scoring a six on the throw of an ordinary six-sided die.

An event

1 2 3 4 5 6

A

s

AA1

Probability of an event

A coin is tossed twice and we are interested in the event (A) that give the same result.

Example

Solution

Sample space =

HH, HT, TH, TT

Event A = (HH, TT)

P(A) =

2/4 = ½

Note: 0 P(A) 1

Example 1

The possibility space consists of the integers from 1 to 25 inclusive.

A is the event ‘the number is a multiple of 5’. is the event ‘the number is a multiple of 3’.

An integer is picked at random.

Find (a) P(A), (b) P(B1)

Solution

)(

)()(

sn

AnAP

Possibility space n(s) = 25

(a)

Number of outcomes in event A n(A) = 5 (5, 10, 15, 20 and 25)

= 5/25 = 1/5

(b)

P(B1) = 1 – P(B) =

1 - 8/25 = 17/25

Example 2

A cubicle die, number 1 to 6, is weighted so that a six is

three times as likely to occur as any other number.

Find the probability of

(a) a six accurring,

(b) an even number occurring.

SolutionPossibility space n(s) = {1, 2, 3, 4, 5, 6, 6, 6, } = n(s) = 8(a)

P( a six) = 3/8

(b)

P( an even number) =

5/8

The pie chart shows the results of 120 people.

A, 240

B, 990

C

D, 1200

E, 420

Example: A car manufacturer carried out a survey in which people were asked which factor from the following list influenced them when buying a car:

A: Colour

B: Service cost

C: Safety

D: Fuel economy

E: ExtrasThe names of those who took part were then placed in a prize draw.

Find the probability that someone who said safety will win the prize.

The pie chart shows the results of 120 people.

A, 240

B, 990

C

D, 1200

E, 420

Solution:

C = 360 – ( 24 + 99 + 120 + 42 ) = 75

P(C) = 75/360 = 25/120 = 5/24