probability students copy

8/11/2019 Probability Students Copy

1/26

1


2/26


3/26

3

Probability A measure of the likelihood that something

happens or the chance that an event can occur.

Probability is measured on a scale from 0 to 1 andmost events therefore have probabilities betweenthese extremes.

Examples are :

Probability of obtaining an 8 on a die,Probability of the next baby to be born at RIPAShospital will be a boy,Probability of obtaining a score of 1,2,3,4,5 or 6on the die.


4/26

Each possible outcome of an experiment iscalled the sample point.

The set of all possible outcomes is thesample space, S.

Example : Toss a dice onceSince S = {1,2,3,4,5,6}n(S) = 6.

4

Sample Space (S)


5/26

5

If a trial has a set of equally likely outcomes,S, then the probability that an event A

occurs is given by P(A) =

i) Its value is between 0 and 1

ii) If P(A) = 0, an impossible event

iii) If P(A) = 1, an event is sure to happen

)()(

S N

A N

Definition of Probability


6/26

Venn diagramsVenn diagrams can be used to represent probabilities.

The outcomes thatsatisfy event A can be

represented by a circle. A

The outcomes that satisfyevent B can be represented

by another circle.

B

The circles can be overlapped to represent

outcomes that satisfy both events.


7/26

a) Complementary event :

If the probability of an event A happening is P(A),then the probability that it does not happen isP(A).

7

Rules of Probability

P(A) + P(A)= 1

Venn diagram

representingComplementaryevents

A

A


8/268

b) Combined events :

P(A B) = P(A) + P(B) P(A B)

This symbol means union or OR

This symbol means intersect or AND

Venn diagrams can

help you to visualizeprobabilitycalculations.


9/26

Example1 Combined Events

A coin and a die are thrown together. Draw asample space diagram and find the probability ofobtaining

a) a headb) a number greater than 4c) a head and a number greater than 4

d) a head or a number greater than 4

9


10/26

Example 2 - Combined Events

The probability the student may pass mathematics is and the probability he will pass Physics is . If the

probability that he will pass both subjects is , what isthe probability that he will pass at least one subject?


11/26

c) Mutually Exclusive eventsIf events A and B are mutually exclusive it implies thatthey cannot occur at the same time.

For example, if a card is picked at random from astandard pack of 52 cards, the events the card is a club and the card is a diamond are mutually exclusive.

However the events the card is a club and the card is aqueen are not mutually exclusive.

If A and B are mutually exclusive, then

11

P(A B) = P(A) + P(B)

In Venn diagramsrepresenting mutuallyexclusive events, thecircles do not overlap.

A B


12/2613


13/26

Example 1 - ( Mutually Exclusive events)

The probability of getting a 5 or 6 whenthrowing a dice is simply

Example 2 - ( Mutually Exclusive events)

What is the probability of throwing a dice andgetting odd numbers ?

13


14/26

d) Independent EventsTwo events are said to be independent if the

occurrence of one has no effect on the probabilityof the second occuring.

14

Example 1 - ( Independent Events)

A coin is tossed and a dice is rolled. What is the

probability of getting a head on the coin and a fouron the dice ?

P(A and B) = P(A) x P(B)


15/26

15

Example 2 : (Independent Events) A and B are independent events. P( A ) = 0.7 and

P( B ) = 0.4.

a) Find P( A B ).

b) Find P( A B ).


16/26

Example 3 - (Independent Events)

What is the probability of getting two 6s? We can enumerate all the possible outcomes in a table :

1,1 2,1 3,1 4,1 5,1 6,1

1,2 2,2 3,2 4,2 5,2 6,2

1,3 2,3 3,3 4,3 5,3 6,3

1,4 2,4 3,4 4,4 5,4 6,4

1,5 2,5 3,5 4,5 5,5 6,5

1,6 2,6 3,6 4,6 5,6 6,6

16


17/26

Tree diagrams are sometimes a useful way offinding probabilities that involve a succession ofevents.

It shows all the possible events. The first event isrepresented by a dot. From the dot, branches aredrawn to represent all possible outcomes of the

event. The probability of each outcome is written onits branch.

We just multiply the probabilities as we go along the

branches to get the required probability.

Note:check that the sum of the probabilities on eachbranch is 1. 17


18/26

Example 1 : (Independent Events using Tree Diagrams)

A bag contains 6 green counters and 4 blue counters. Acounter is chosen at random from the bag and then replaced.

This is repeated two more times.Find the probability that the 3 counters chosen are

a) all green

b) not all the same colour.


19/26

Example 2: (Independent Events using Tree Diagrams)

A coin is flipped 3 times. What is theprobability of :

a) getting three heads.

b) getting two heads and one tail,(in the order): P(H,H,T).

c) getting one head and two tails,

(in the order): P(H,T,T).

d) getting two heads and one tail in anyorder.

19


20/26

FIRST SECOND THIRD

H H HH H T

H T H

H T T

T H H

T H T

T T H

T T T

20


21/26

Example 3 - (Independent Events using Tree Diagrams)

A firm is independently working on twoseparate jobs. There is a probability of only0.3 that either of the jobs will be finished ontime.

a) Construct a probability tree of the problem.

b) Calculate the probability that :i) Both of the jobs are finished on timeii) At least one of the jobs is finished on

time 21


22/26

Example 4 : (Independent Events using Tree Diagrams)

A vacuum cleaner salesmen MJ must make twocalls per day, one in the morning and one in theafternoon. MJ has probability of 0.4 of selling acleaner on any call. The morning and afternoon

results are independent of each other. Find theprobability that, in one day :

a) MJ sells just one cleaner.b) MJ doesnt sell any cleaner.

22


23/26

23

)()(

)|( B P

B A P B A P

e. Conditional Probability


24/26

24

Example 1 Conditional Probability

Ali travels to work either by bus or by taxi or by

bicycle. The probability that he travels by bus is 0.3,by taxi is 0.5 and by bicycle is 0.2. The probabilitythat he arrives on time is 0.6 if he travels by bus, 0.9if he travels by taxi and 0.8 if he travels by bicycle.

a) Draw a tree diagram to show this information.

b) Find the probability that Ali does not arrive at work

on time.c) Find the conditional probability that Ali traveled by

bus , given that he does not arrive at work ontime.


25/26

Example 2 : (Conditional Probability)

A firm has tendered for two independent contracts. It

estimates that it has probability 0.4 of obtainingcontract E and probability of 0.1 obtaining contractF.a) Find the probability that the firm obtains exactly

one contract.

b) Given that the firm obtained exactly one contract,

find the probability that it is from contract F.

25


26/26

26

References

Lecture & Tutorial Notes from Department of Business & Management,Institute Technology Brunei, Brunei Darussalam.

ACT Education Solutions (2005) Global Assessment Certificate Student Manual, ACT Education Solutions Limited, Australia

MDIS (2010) Diploma in Business Management Course Study Booklet,Management Development Institute of Singapore Pte Ltd, Singapore.General Studies Department (2012), Introduction to Statistics : LectureNotes & Slides, Brunei Polytechnic, Brunei Darussalam

Phua, S. et al (1991). Statistics for Business (2nd Edition). FT Law & TaxAsia Pacific : Singapore

Weiers , R.M. (2008) Introduction to Business Statistics (6thEdition). Thompson South-Western : Canada

probability students copy

Documents