**Unformatted text preview: **Chapter 2: Probability
I calculated the probability of me passing the class and the answer I got
was “banana”… Things do not look good. 1 2.1 Sample spaces and events
Experiment (in a wide sense): Any activity whose outcome is subject to uncertainty.
• E.g.: Coin toss, roll dice, life span of equipment, number of errors when compiling a
program for the first time, counting white blood cells, obtaining blood type.
Sample space of an experiment is the set of all possible outcomes of the experiment
• 1 coin toss = {, }
• White blood cell count = [0, ]
• # errors when compiling a program = 0, 1, 2, 3, … , +∞
Events: An event is a collection (subset) of outcomes. An event is a simple event if it is a
single outcome, and a compound event if it consists of more than one outcomes. 2 2.1 Sample spaces and events
Example (1): A fuse could be defective (D) or not defective (N). We examine 3 fuses. What are
the possible outcomes? = {, , , , , , , } (8 outcomes)
• Event : The first fuse is defective, the other two are not. = {} • Event : There is exactly 1 defective fuse. = {, , } • Event : There are at least 2 defective fuses. = {, , , } Infinite number of outcomes = infinite number of simple events
= infinite number of events
3 2.1 Sample spaces and events
Example (2): Compiling a program
Event : At most 3 errors.
Event : An even number of errors. = {0,1, 2, 3} = {0, 2, 4, 6, … } SET THEORY
1. Complement of an event is the set of all outcomes in that are not in .
Denoted by ′ or .
2. Union of events , ( ∪ , “ or ”) is the event consisting of all outcomes that
are either in or in , or in both. 3. Intersection of events , ( ∩ , “ and ”) is the event consisting of outcomes
that are both in and in .
4 2.1 Sample spaces and events
Example (3): A factory has 6 production lines in total. The number of product lines that
are operational at any given time varies (is subject to uncertainty).
Sample space = 0, 1, 2, 3, 4, 5, 6
Event = 0, 1, 2, 3, 4
Event = 3, 4, 5, 6 Event = 1, 3, 5 Then:
•
•
•
•
•
• ′ =
∪ =
∪ =
∩ =
∩ =
( ∩ )′ =
5 2.1 Sample spaces and events
Venn diagrams (graphical representation of set theory concepts) ∅ null event (no outcome/empty event)
• If ∩ = ∅, then and are called mutually exclusive or disjoint events
• Simple events are always disjoint
6 2.2 Axioms, interpretations and properties of probabilities
Event → () ≡ The likelihood that will occur (Probability of ) Axioms
1. ≥ 0
2. = 1
3. If 1 , 2 , … an infinite collection of disjoint events, then
∞ 1 ∪ 2 ∪ ⋯ = ራ =1 ∞ = =1
7 2.2 Axioms, interpretations and properties of probabilities
Proposition ∅ =0
Proof: From Axiom 3. If = ∅ for all , then
∞ ∅ ∪ ∅∪ ⋯ = ∅ = ∅ → ∅ = 0
=1 Proposition: If disjoint =ڂ1 = σ=1 Proof: From Axiom 3. Set = ∅ for all > , then ራ =1 ∞ ∞ ∞ ∞ = ራ = = + = + ∅ = =1 =1 =1 =+1 =1 =+1 =1 8 2.2 Axioms, interpretations and properties of probabilities
Example: Coin toss = {, } =∪ = ∪ = 1 → + = 1 → = 1 − () Proposition
′ = 1 − () Proof: = ∪ ′ → = ∪ ′ = 1 → + ′ = 1 → ′ = 1 − ()
9 2.2 Axioms, interpretations and properties of probabilities
Example: Accidents in a factory
Shift Day
Night Human error
10%
13% Reason
Equipment malfunction
35%
42% = {, , , } with = 0.1, = 0.13, = 0.35, = 0.42 ℎ = ℎ ℎ = 10 2.2 Axioms, interpretations and properties of probabilities
Interpreting probabilities
Assume an experiment that is performed times under identical conditions.
• : an event
• (): the number of times occurred.
• ()/ is the relative frequency of occurrence of For “small” , the value ()/ will “fluctuate”
As gets “large” (going to infinity – “on the long run”) ()/ will converge to a single
value ≡ ()
E.g. (a given machine will malfunction during this year) = 0.1 → 10% of ALL the
machines of such type will malfunction
Objective vs subjective interpretation of probability (does “long run” really exist?)
11 2.2 Axioms, interpretations and properties of probabilities
Probability properties Events and . Then ∪ = + − ( ∩ )
Proof: ∪ = + ∩ ′ = + − ∩ + − ( ∩ ) = 12 2.2 Axioms, interpretations and properties of probabilities
Probability properties ∪∪ =
= + + − ∩ − ∩ − ∩ + ∩∩ ∪∪∪ =⋯ 13 2.2 Axioms, interpretations and properties of probabilities
Probability properties ∪ ∩ = ∩ + ∩ − ∩∩ 14 2.2 Axioms, interpretations and properties of probabilities
Determining probabilities systematically • Sample space with simple events 1 , 2 , 3 , …
• are disjoint and σ∞
=1 = 1
An event can be represented as = ራ then = σ ∈ 15 2.2 Axioms, interpretations and properties of probabilities
Example: Finite number () and equally likely outcomes , = 1,2, … , , with = =1 =1 1 = = = = 1 → = An event is the union of () in number equally likely to happen simple events
(outcomes). Then 1 # = = =
= ∈ ∈ 16 2.2 Axioms, interpretations and properties of probabilities
NOTE: You can think of probabilities as the areas in the Venn diagrams
over the total area of the sample space = NOTE: Oftentimes it is very useful to think of probabilities as percentages.
((ℎ ℎ)= percentage of people that have black hair)
17 2.3 Counting techniques
Estimating probabilities for equally likely outcomes is quite straightforward. The difficulty lies
in counting the number of favorable outcomes (here is where counting techniques come
useful)
Product rule for ordered pairs (, ) ≠ (, ) In an ordered pair (, ) where there are 1 possibilities for and 2 possibilities for , the
number of pairs is 1 ∗ 2 .
Product rule for tuples
An ordered pair of objects is called a -tuple
If there are possibilities (choices) for the -th object in a -tuple, then there are
1 ∗ 2 ∗ ⋯ ∗ possible -tuples. (This is a generalization of the case of ordered pairs) 18 2.3 Counting techniques
Tree diagrams
1 , 2 , 3 , 4 “treatments”, 1 , 2 , 3 “populations”. = 4 ∗ 3 = 12 pairs 19 2.3 Counting techniques
Permutations and Combinations
There are individuals. I want to select a subset of size .
Question: Does the order of selection matter?
Yes – ordered subset: Permutations. # of permutations , No – unordered subset: Combinations. # of combinations , , or
choose ” “ 20 2.3 Counting techniques
Example: : Selecting = 4 students out of = 40
Candidates: Yiannis, Hubert, Carl, Bob, Katie,…
Positions: President, Vice President, Treasurer, Secretary
President
Y
H
H
… Vice president
H
Y
Y
… Treasurer
C
C
C
… Secretery
B
B
K
… How many possible sets do I have?
There are 40 candidates for president. After the selection of the president I have 39
candidates for the VP, and so on. In total we have (product rule for tuples):
40 ∗ 39 ∗ 38 ∗ 37 sets
• We can write this as: 40 ∗ 39 ∗ 38 ∗ 37 ∗ 36∗35∗34∗⋯∗2∗1
36∗35∗34∗⋯∗2∗1 = 40!
36! = 40!
40−4 !
21 2.3 Counting techniques
Permutations of objects by , !
=
− ! Permutations of objects by : , = !
− ! , = Combinations of objects by : , =
=
,
(we “group” different orderings together)
, !
0 ! = ! !
− ! ! !
=
! − !
22 2.3 Counting techniques
Example: 100 songs on my iPod. 10 of them are Beatles songs. I have my iPod
on shuffle. What is the probability that the 5th song played is the 1st Beatles
song? = Number of ways to select the first 5 songs: 100 ∗ 99 ∗ 98 ∗ 97 ∗ 96 = 5,100
Number of ways to select the first 4 songs if they are not Beatles songs: = 4,90
Number of ways to select the 5th song if it is a Beatles song: = 10
4,90 ∗ 10
90 ∗ 89 ∗ 88 ∗ 87 ∗ 10 =
=
= 0.06
5,100
100 ∗ 99 ∗ 98 ∗ 97 ∗ 96
23 2.3 Counting techniques
Example: 15 Apple and 10 Dell computers are bought. 6 of them will be given to
students.
a) What is the probability that exactly 3 will be Apple?
3 =exactly 3 of the 6 are Apple
Total # of outcomes: 25
6 # of favorable outcomes
computers) = 177100
15
3 ∗ 10
3 3 = 54600 (the students get 3 Apple and 3 Dell 54600
=
= 0.3083
177100
24 2.3 Counting techniques
Example (cont.)
b) What is the probability that at least 3 will be Apple?
Define 4 , 5 , 6 similarly to 3 . Then: 3 = 3 ∪ 4 ∪ 5 ∪ 6 = NOTE: Oftentimes it is much easier to work with probabilities if we define them in the context
of a “variable”. : number of Apple computers given to students 3 = = 3 , 3 = ( ≥ 3)
25 2.4 Conditional probability (|)
The occurrence of an event may affect the probability of an event occurring
: a person has a disease : the blood test was negative () vs. (|)
Example: 2 assembly lines 1 , 2
1 produced 8 components with 2 being defective
2 produced 10 components with 1 being defective Line 1
Line 2 Defective
2
1 Not defective
6
9 We select one component randomly
Event : line 1 component was selected
Event : the component was defective ∩ = 8
= 0.44
18
3
() = = 0.17
18
2 = =
3 () = 2
18 0.66 26 2.4 Conditional probability (|)
Conditional probability of , given that has occurred is: ∩ = as long as () ≠ 0.
Note: ≠ Given that happened, the sample space has changed from to .
So the probability of is now: ∩ ∩ / ∩ =
=
= / 27 2.4 Conditional probability
Example:
Probability of liking cats = 0.5
Probability of liking dogs = 0.6
Probability of liking cats AND dogs ∩ = 0.4
Then = 28 2.4 Conditional probability
Example: = 0.14, = 0.23, = 0.37 ∩ = 0.08, ∩ = 0.09, ∩ = 0.13 ∩ ∩ = 0.05 (a) What is | ? ∩
0.09 | =
=
= 0.2432 0.37 29 2.4 Conditional probability
Example (cont.):
b) What is | ∪ ?
Hint: Define event = ∪ 30 2.4 Conditional probability
Example (cont.):
c) ∪ | = d) | ∪ ∪ = 31 2.4 Conditional probability (|)
Multiplication rule for ∩ ∩ = = (|)
Useful when events are different stages in succession 1 ∩ 2 ∩ 3 = 3 1 ∩ 2 )(1 ∩ 2 )
= 3 1 ∩ 2 )(2 |1 )(1 ) 32 2.4 Conditional probability
Bayes’ Theorem preliminaries
Exhaustive events: , = 1, … , : =ڂ1 = Law of total probability
If exhaustive and mutually exclusive events, then for any other event = 1 1 + 2 2 + ⋯ + = =1 = ( ∩ )
=1
33 2.4 Conditional probability
Example: I have 3 email accounts (1 , 2 , 3 )
• 70% of my emails come to 1 . 1% of the emails in 1 are spam
• 20% of my emails come to 2 . 2% of the emails in 2 are spam
• 10% of my emails come to 3 . 5% of the emails in 3 are spam What is the probability that an email I get is spam? :{email in account },
: {email is spam} = 34 2.4 Conditional probability
BAYES’ THEOREM , = 1, 2, … , exhaustive and mutually exclusive events. Then for any
event ∩ | =
= ,
()
σ=1 = 1,2, … , • is the prior probability of event • | is the posterior probability of event (after taking into
account the information of )
35 2.4 Conditional probability
Example (rare diseases):
1 in 1000 people has the exploding head syndrome. A company has developed a diagnostic test.
We know that if an individual has the syndrome, a positive result on the test has probability
99%. If the individual does not have the syndrome, a positive result has probability 2%. What is
the probability of someone having the syndrome if the results are positive?
Events 1 = {an individual has the syndrome}, 2 = 1 ′, = {the diagnostic test gave a positive
result} ℎ ℎ ℎ = 1 1 =
36 2.5 Independence
Two events are independent if (|) = (). Otherwise they are
dependent.
• Also = ∩
() = () = ()()
() = • Also, if and are independent, then and ′ are independent
• Also, if and disjoint (mutually exclusive), then =
(i.e. they are dependent). ∩
() = 0 = 0, • Also, if and are independent, then ∩ = = (multiplication rule for independent events)
37 2.5 Independence
Examples: a) 2 dice rolls
: { 1 ℎ }
: { 5 ℎ }
b)
: { ℎ }
: { ℎ }
More than 2 events?
Events , = 1,2, … , are mutually independent if for every = 2,3, … , , and every
subset of indices 1 , 2 , … , 1 ∩ 2 ∩ ⋯ ∩ = 1 2 … 38 2.5 Independence
Example: Consider the following system consisting of 2 subsystems
(subsystem 1-2-3, subsystem 4-5-6).
• Assume that for the system to work, at least one of the two subsystems
should work.
• Assume that each individual component works with the same probability
of 0.9.
• Assume that the individual
components are independent.
• What is the probability that the
system works
39 2.5 Independence
Example (cont.):
Event : {component works}, with = 0.9 = = 1 ∩ 2 ∩ 3 ∪ 4 ∩ 5 ∩ 6
= 1 ∩ 2 ∩ 3 + 4 ∩ 5 ∩ 6 − 1 ∩ 2 ∩ 3 ∩ 4 ∩ 5 ∩ 6
=∗∗+∗∗−∗∗∗∗∗
= 0.93 + 0.93 − 0.96 = 0.927
(Conversely, if there was only one subsystem, the probability would be 0.93 = 0.73)
40 2.5 Independence
Example: Assume that for the system to work, BOTH the subsystems should
work.
Event : {component works}, with = 0.9 = = 1 ∪ 2 ∩ 3 ∩ 4
41 2.5 Independence
Example (cont.): Hint: 1 ∪ 2 ∩ 3 ∩ 4 = 1 ∪ 2 ∩ = 42 ...

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