PROBABILITY - Text version of PowerPoint slides (minus the graphics!)

Slide 1

PROBABILITY
EMPIRICAL:
relationship or expectation is found
by experiment or use of historical
data
THEORETICAL
expectation is found by use of logic,
symmetry or listing outcomes
SUBJECTIVE
based on belief or judgement of
what will happen

Slide 2

Some basics....... 

EXCLUSIVE AND NON-
EXCLUSIVE EVENTS 
Two events are mutually exclusive if
they are not related
Two events are not exclusive or joint
if they can occur together
Examples:
throwing a 1 or 6
picking a Heart or a Picture card

Slide 3

RULES OF ADDITION 
If events are exclusive: 
Prob (A or B) = P(A u B) 
= P(A) + P(B) 
P( 1 or 6 ) = P (K or Q) = 

Slide 4

If events are not exclusive:
P( A u B) = P(A) + P(B) - P(A n B)
P(Heart or Queen) =
P(Spade or Picture) =
Investigate throwing two dice and 
adding the totals: 
How many possible outcomes are 
there? 
6 . . . . . . 
5 . . . . . . 
4 . . . . . . 
3 . . . . . . 
2 . . . . . . 
1 . . . . . . 
1 2 3 4 5 6 

Find the probabilities of 
throwing: 
• a double 
• a total of 10 
• a double or a total of 10 
• at least one 6 
A picture or listing outcomes helps 

Slide 5

INDEPENDENT, DEPENDENT 
AND CONDITIONAL 
PROBABILITIES 
Two events are INDEPENDENT if 
the occurrence of one is not affected 
by the occurrence of the other one 
example: throwing a die 
If an event depends on or is affected 
by what has happened before then 
the events are DEPENDENT or the 
second event is CONDITIONAL on 
the first 
example: passing an exam second go 

Slide 6

MULTIPLICATIVE RULES 
If events are independent 
P(A n B) = P(A) . P(B) 
e.g: P( two sixes) = P(6) . P(6) 
If events are dependent 
P( B/ A) = P( A n B) 
P(A) 
e.g: 
P(Heart) 
B/A means B 
once A has 
happened 
P(<4 / Heart ) = P(<4 and Heart) 

Slide 7

TREE DIAGRAMS 
3 white and 7 blue balls in a bag 
1. If a ball is selected and then replaced... 
w 
b 
w 
b 
w 
b
Events are independent 

Slide 8

TREE DIAGRAMS 
3 white and 7 blue balls in a bag 
1. If a ball is selected and then replaced... 
w 
b 
w 
b 
w 
bEvents are independent 
3/10 
7/10 
3/10 
7/10 
3/10 
7/10 
WW = 3/10 x 3/10 = 9/100 


Slide 9

2. If a ball is selected but not replaced... 
w 
b 
w 
b 
w 
b
Events are dependent 
3/10 
7/10 
2/9 
7/9 
3/9 
6/9 
WW = 3/10 x 2/9 = 6/90 

Slide 10

Age Male Female totals 
< 30 100 75 175 
30+ 50 25 75 
totals 150 100 250 
MARGINAL 
PROBABILITIES 
are found on edge of 
table 
P(F) = 100/250 = 0.4JOINT PROBABILITIES 
involve two classifications 
P( M and 30+) = 50/250 = 0.2 
The results of a survey of 250 customers at a jeans store 
can help us in marketing: 
5

Age Male Female totals 
< 30 100 75 175 
30+ 50 25 75 
totals 150 100 250 

Slide 11

CONDITIONAL PROBABILITIES 
a probability once another condition 
has occurred 
P(F / <30) = 75/175 = 0.429 
Condition 
<30 

Slide 12


Find the percentage or probability give info 
about the customers and underlying trends: 
age or gender profile e.g. P(30+) P(F) 
conditional probabilities look for a pattern: 
age versus gender 
e.g. P(30+/ F) and P(30+/M)
or P(F/<30) and P(M/<30) 
Age Male Female totals 
< 30 100 75 175 
30+ 50 25 75 
totals 150 100 250 

Slide 13

BAYESIAN PROBABILITY 
Thomas Bayes - Minister 1702 - 61 
· He found a way to estimated the 
probability of an event that had 
already happened by using 
information from a sample 
PRIOR PROBABILITY 
based on historical data 
POSTERIOR PROBABILITY 
uses historical data and new information 
from surveys, testing etc. 

Slide 14

The formula is 
P( A / B ) = P(A). P (B / A) 
P(A). P(B / A) + P(A'). P(B / A') 
Do not panic! 
The formula you find in text books look 
complicated but using a tree diagram is 
straightforward and gives the same answer! 
after B has 
happened happened 
Prob of A Prob A * conditional 
prob of B after A has 
Total prob of B 


Slide 15

A Manufacturing process needs to meet specifications. 
When it does the process is In Control. 
· If it is in control the proportion of defectives is 0.05 
· If it is out of control the proportion of defectives is 
0.20 
· Historical data suggests it is in control 90% of the time 
We can up-date this historical info using Bayes..... 
c 
nc 
nd 
d 
nd 
d 
.9 
.1 
.95 
.05 
.8 
.2 
=0.855 
P(ND/C) 
=0.08 
=0.045 
=0.02 
0.935 
0.065 
ND 
D 

Slide 16

Conditional probs 

the process being in control is: 
P(C/ND) = 0.855/0.935 = 0.914 i.e. 91.4% 

Slide 17

Sampling: 
If you pick out a defective item, the probability of the 
process being in control is: 
P(C/D) = 0.045/0.068 = 0.662 i.e. 66.2% 
If you pick out a non-defective item, the probability of 
The Prior probability of 0.9 is up-dated according to sample state 

Slide 18

Analyse the following data using different probabilities. 
What is the relationship between age, dress and buying 
behaviour? 
A probability 
case study based 
on 
actual data in 
USA 
· 12 Up - market fashion stores 
selling women clothing. 
· These attract many window 
shoppers and tourists. 
· It would be useful if staff could 
identify serious buyers. 
· The M.R. thinks that buying 
pattern is affected by age and dress 
of the shoppers. 

Slide 19

· Data is collected recording the 
behaviour of a random selection of 
shoppers in one store. 
Data One: under 40 
Female buyers 40 plus 
well buyer 2 8 
dressed 
non16 
14 
buyer 
casually buyer 34 6 
dressed 
non50 
70 
buyer 
Slide 20


Sub-totals are useful.... 
Data One: Female under 40 plus total 
buyers 40 
well dressed buyer 2 8 10 
non16 
14 30 
buyer 
18 22 40 
casually buyer 34 6 40 
dressed 
non50 
70 120 
buyer 
84 76 160 
102 98 200 
10

Slide 21

The following information comes from the same store, 
but for male buyers. 
Analyse the data using a tree diagram and Bayes. 
Who should the shop assistant target? 
Male buyers 
· form 1/3 rd of customers 
· 6 out of 10 made a purchase 
· of those who made a purchase 2 out 
of 10 wore suits 
· of those who didn't make a purchase 
9 out of 10 were not in a suit 

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