STAT: Random experiments(2)

1. What have we learned so far?What have we learned so far?
POPULATION
SAMPLE
• We describe our sample using measurements and data presentation tools
• We study the sample to make inferences about the population
• What permits us to make the inferential jump from sample to population?
FDT, Graphs, Mean,
Median, Mode,
Standard Deviation,
Variance

2. PROBABILITYPROBABILITY
POPULATION
SAMPLE

In probability, we use the In probability, we use the
population information to infer population information to infer
the probable nature of the the probable nature of the
sample.sample.

3. Example: TOSSING A COIN Example: TOSSING A COIN

Suppose a coin is tossed once Suppose a coin is tossed once
and the up face is recordedand the up face is recorded

The result we see and The result we see and
recorded is called an recorded is called an
OBSERVATION or or
MEASUREMENT
The process of making an
observation is called an
EXPERIMENT.

4. Definition:Definition:
RANDOM EXPERIMENTRANDOM EXPERIMENT

Is a Is a processprocess or or procedureprocedure, , repeatable under repeatable under
basically the same condition basically the same condition (this repetition (this repetition
is commonly called a is commonly called a TRIALTRIAL)), leading to , leading to well-well-
defined outcomesdefined outcomes..

It is It is randomrandom because we can never tell in because we can never tell in
advance what the outcome/realization is going advance what the outcome/realization is going
to be, even if we can specify what the possible to be, even if we can specify what the possible
outcomes are.outcomes are.

5. Example: TOSSING A DIEExample: TOSSING A DIE

Consider the simple Consider the simple
random experiment of random experiment of
tossing a die and tossing a die and
observing the number observing the number
on the up face.on the up face.

There are six There are six basic
possible outcomes to to
this random experiment.this random experiment.
1.1.Observe a Observe a 11
2.2.Observe a Observe a 22
3.3.Observe aObserve a 33
4.4.Observe a Observe a 44
5.5.Observe a Observe a 55
6.6.Observe a Observe a 66

6. Definitions:Definitions:
SAMPLE POINT & SAMPLE SPACESAMPLE POINT & SAMPLE SPACE

A A SAMPLE POINT is the most basic is the most basic
outcome of a random experiment.outcome of a random experiment.

The The SAMPLE SPACE is the set of all is the set of all
possible outcomes of a random possible outcomes of a random
experiment. It is experiment. It is denoted by the Greek denoted by the Greek
letter omega (letter omega (ΩΩ) or S) or S. This is also known . This is also known
as the as the universal setuniversal set. .

7. Examples:Examples:

Sample space of the “Tossing of Coin” Sample space of the “Tossing of Coin”
experiment:experiment:
ΩΩ = = {Head, Tail}{Head, Tail}

Sample spaceSample space of the “Tossing of Die” of the “Tossing of Die”
experiment:experiment:
ΩΩ = = {1, 2, 3, 4, 5, 6}{1, 2, 3, 4, 5, 6}

8. Exercise 1:Exercise 1:
1.1.Two coins are tossed, and their up faces are Two coins are tossed, and their up faces are
recorded. What is the sample space for this recorded. What is the sample space for this
experiment?experiment?
Coin 1Coin 1Coin 2Coin 2
HeadHeadHeadHead
TailTailHeadHead
HeadHead TailTail
Tail Tail TailTail
ΩΩ = {HH, TH, HT, TT} = {HH, TH, HT, TT}

9. Exercise 2:Exercise 2:
2.2.Suppose a pair of dice is tossed . What is the sample Suppose a pair of dice is tossed . What is the sample
space for this experiment?space for this experiment?
ΩΩ = {1-1,1-2,1-3,1-4,1-5,1-6, = {1-1,1-2,1-3,1-4,1-5,1-6,
2-1,2-2,2-3,2-4,2-5,2-6,2-1,2-2,2-3,2-4,2-5,2-6,
3-1,3-2,3-3,3-4,3-5,3-6,3-1,3-2,3-3,3-4,3-5,3-6,
4-1,4-2,4-3,4-4,4-5,4-6,4-1,4-2,4-3,4-4,4-5,4-6,
5-1,5-2,5-3,5-4,5-5,5-6,5-1,5-2,5-3,5-4,5-5,5-6,
6-1,6-2,6-3,6-4,6-5,6-6}6-1,6-2,6-3,6-4,6-5,6-6}

10. Exercise 3:Exercise 3:
3.3.A sociologist wants to determine the gender of A sociologist wants to determine the gender of
the first two children of families with at least two the first two children of families with at least two
(2) children in a baranggay in Dasmarinas, (2) children in a baranggay in Dasmarinas,
Cavite. He then observes and records the gender Cavite. He then observes and records the gender
of the first 2 children of these families.of the first 2 children of these families.
ΩΩ = = {MM, MF, FM, FF}{MM, MF, FM, FF}
where, M represents Male and F represents Female

11. Exercise 4:Exercise 4:
4.4.Consider the experiment of recording the Consider the experiment of recording the
number of customers placing their order at number of customers placing their order at
the “Drive Thru” of a particular McDonald’s the “Drive Thru” of a particular McDonald’s
branch per day. What is the sample space for branch per day. What is the sample space for
this random experiment?this random experiment?
ΩΩ = = {0, 1, 2, 3, 4, 5, …}{0, 1, 2, 3, 4, 5, …}

12. Exercise 5:Exercise 5:
5.5.Suppose GMA Foundation wanted to know the Suppose GMA Foundation wanted to know the
effectiveness of their feeding program in a effectiveness of their feeding program in a
particular baranggay in Dasmarinas. The particular baranggay in Dasmarinas. The
coordinator records the change in the children’s coordinator records the change in the children’s
weight to height ratio (BMI). What is the sample weight to height ratio (BMI). What is the sample
space for this random experiment?space for this random experiment?
ΩΩ = = {y / y {y / y ≥ 0≥ 0 } }
where, y = the change in a child’s BMI, assuming it
is not possible for a child to have a decrease in BMI while
enrolled in the feeding program.

13. Types of Sample Spaces:Types of Sample Spaces:
1.1.FINITE SAMPLE SPACEFINITE SAMPLE SPACE

Is a sample space with Is a sample space with finite numberfinite number of possible of possible
outcomes (sample points).outcomes (sample points).

Exercises 1 to 3Exercises 1 to 3 are examples of finite sample spaces. are examples of finite sample spaces.
1.1.INFINITE SAMPLE SPACEINFINITE SAMPLE SPACE

Is a sample with Is a sample with infinite numberinfinite number of possible outcomes. of possible outcomes.

Exercise 4Exercise 4 is an example of a is an example of a countablecountable infinite infinite
sample space.sample space.

Exercise 5Exercise 5 is an example of a is an example of a uncountableuncountable infinite infinite
sample spacesample space..

14. Natures of Sample SpacesNatures of Sample Spaces
1.1.DISCRETE SAMPLE SPACE DISCRETE SAMPLE SPACE

Is a sample space with a Is a sample space with a countable (finite or countable (finite or
infinite) number of possible outcomesinfinite) number of possible outcomes..

Examples are Examples are Exercises 1 to 4Exercises 1 to 4
1.1.CONTINUOUS SAMPLE SPACECONTINUOUS SAMPLE SPACE

Is a sample space with a Is a sample space with a continuum of possible continuum of possible
outcomesoutcomes..

Example is Example is Exercise 5Exercise 5..

15. Recall the “Tossing of Die” experiment.Recall the “Tossing of Die” experiment.
Suppose we are interested in the
outcome that an even number will
come up.
1
5
3
2
4 6
A
ΩΩ
Let EVENT A, be
the collection of
sample points that
fulfill the outcome
we are interested in,
i.e., an even number
will come up.

16. Definition: EVENTDefinition: EVENT

An An EVENT EVENT is a is a subset of the sample spacesubset of the sample space..

It is denoted by any letter of the English It is denoted by any letter of the English
alphabet.alphabet.

An event is an An event is an outcome of a random outcome of a random
experimentexperiment..

An event is a An event is a specific collection of sample specific collection of sample
pointspoints..

17. Examples:Examples:
1.1.ΩΩ = = {Head, Tail}{Head, Tail}
Let A = Let A = {Head}{Head}, , the event of a Head turning up.
Let B =Let B = {Tail} {Tail}, , the event of a Tail turning up.
2.2.ΩΩ = = {{Head-Head, Head-Tail, Tail-Head, Tail-Tail}Head-Head, Head-Tail, Tail-Head, Tail-Tail}
Let X = Let X = {Head-Head, Head-Tail, Tail-Head}{Head-Head, Head-Tail, Tail-Head}, ,
the event of at least one Head will turn up.
Let Y =Let Y = {Tail-Tail, Tail-Head, Head-Tail} {Tail-Tail, Tail-Head, Head-Tail}, ,
the event of a at least one Tail will turn up.

18. Exercise 6:Exercise 6:
Given the sample space Given the sample space
ΩΩ, , for the single toss for the single toss
of a pair of fair dice, of a pair of fair dice,
list the elements of list the elements of
the following events:the following events:

AA = event of = event of
obtaining a sum that obtaining a sum that
is an is an even numbereven number..

BB = event of obtaining = event of obtaining
a sum that is an a sum that is an odd odd
number.number.
ΩΩ = {1-1,1-2,1-3,1-4,1-5,1-6, = {1-1,1-2,1-3,1-4,1-5,1-6,
2-1,2-2,2-3,2-4,2-5,2-6,2-1,2-2,2-3,2-4,2-5,2-6,
3-1,3-2,3-3,3-4,3-5,3-6,3-1,3-2,3-3,3-4,3-5,3-6,
4-1,4-2,4-3,4-4,4-5,4-6,4-1,4-2,4-3,4-4,4-5,4-6,
5-1,5-2,5-3,5-4,5-5,5-6,5-1,5-2,5-3,5-4,5-5,5-6,
6-1,6-2,6-3,6-4,6-5,6-6}6-1,6-2,6-3,6-4,6-5,6-6}

19. Exercise 6: (cont.)Exercise 6: (cont.)
ΩΩ = {1-1,1-2,1-3,1-4,1-5,1-6, 2-1,2-2,2-3,2-4,2-5,2-6, = {1-1,1-2,1-3,1-4,1-5,1-6, 2-1,2-2,2-3,2-4,2-5,2-6,
3-1,3-2,3-3,3-4,3-5,3-6, 4-1,4-2,4-3,4-4,4-5,4-6,3-1,3-2,3-3,3-4,3-5,3-6, 4-1,4-2,4-3,4-4,4-5,4-6,
5-1,5-2,5-3,5-4,5-5,5-6, 6-1,6-2,6-3,6-4,6-5,6-6}5-1,5-2,5-3,5-4,5-5,5-6, 6-1,6-2,6-3,6-4,6-5,6-6}
A = {1-1,1-3,1-5,2-2,2-4,2-6,3-1,3-3,3-5,4-2,A = {1-1,1-3,1-5,2-2,2-4,2-6,3-1,3-3,3-5,4-2,
4-4,4-6,5-1,5-3,5-5,6-2,6-4,6-6}4-4,4-6,5-1,5-3,5-5,6-2,6-4,6-6}
B = {1-2,1-4,1-6,2-1,2-3,2-5,3-2,3-4,3-6,4-1,B = {1-2,1-4,1-6,2-1,2-3,2-5,3-2,3-4,3-6,4-1,
4-3,4-5,5-2,5-4,5-6,6-1,6-3,6-5}4-3,4-5,5-2,5-4,5-6,6-1,6-3,6-5}

20. Types of EventsTypes of Events
1.1.ELEMENTARY EVENTELEMENTARY EVENT

An event consisting of An event consisting of ONE possible outcomeONE possible outcome..

Example is the elementary events of Example is the elementary events of
ΩΩ = = {Head, Tail}{Head, Tail}
A A = = {Head} and B {Head} and B = = {Tail}{Tail}
ΩΩ = = {Pass, Fail}{Pass, Fail}
C C = = {Pass} and D {Pass} and D = = {Fail}{Fail}

21. Types of EventsTypes of Events
2.2.IMPOSSIBLE EVENTIMPOSSIBLE EVENT

An event consisting of An event consisting of NO outcomeNO outcome..

Given the sample space of all possible products Given the sample space of all possible products
that can be purchased from a shoe store.that can be purchased from a shoe store.
ΩΩ = = {Sandals, Slippers, Pumps, Moccasins, Rubber {Sandals, Slippers, Pumps, Moccasins, Rubber
Shoes, Bags, Belts, Accessories}Shoes, Bags, Belts, Accessories}
Let A be the event that one can buy a chain saw Let A be the event that one can buy a chain saw
in a shoe store. Thus A in a shoe store. Thus A = = { } or { } or ϕϕ (null)(null)

22. Types of EventsTypes of Events
3.3.SURE EVENTSURE EVENT

An event consisting of An event consisting of ALL the possible outcomesALL the possible outcomes..

Given the “Tossing of a Die” experiment.Given the “Tossing of a Die” experiment.
Let K be the event that a number less than or equal Let K be the event that a number less than or equal
to 6 will occur if a die is thrown.to 6 will occur if a die is thrown.

23. Types of EventsTypes of Events
4.4.COMPLEMENT EVENTCOMPLEMENT EVENT

Is the set of all elements of the sample space Is the set of all elements of the sample space
which are not in the event, A. which are not in the event, A.

Denoted by ADenoted by A
cc
or A or A''

Given the “Tossing of a Die” experiment.Given the “Tossing of a Die” experiment.
ΩΩ = = {1, 2, 3, 4, 5, 6}{1, 2, 3, 4, 5, 6}
If A = If A = {2, 4, 6}, the event that an even number {2, 4, 6}, the event that an even number
will come up,will come up,
Then Then AA
c c
= = {1, 3, 5}{1, 3, 5}

24. Operations On EventsOperations On Events
1.1.INTERSECTION of 2 events A and B, denoted by INTERSECTION of 2 events A and B, denoted by
AA∩B, is the event containing all elements that are ∩B, is the event containing all elements that are
common to events A and B.common to events A and B.
Example:Example:
ΩΩ = = {a, b, c, d, e, f}{a, b, c, d, e, f}
A = {a, b, c, d}A = {a, b, c, d}
B = {c, d, e, f}B = {c, d, e, f}
A ∩ B = {c, d}A ∩ B = {c, d}
ΩΩ ∩ A = {a, b, c, d}∩ A = {a, b, c, d}

25. Definition: Definition:
MUTUALLY EXCLUSIVE EVENTSMUTUALLY EXCLUSIVE EVENTS

Two events are mutually exclusive if they Two events are mutually exclusive if they
cannot both occur simultaneously. cannot both occur simultaneously.

That is, AThat is, A∩B = { } or ∩B = { } or ϕϕ

Example Let C = {1, 2, 3} and D = {a, b, c} Example Let C = {1, 2, 3} and D = {a, b, c}
Then Then CC∩D = { } ∩D = { }

26. Operations On EventsOperations On Events
2.2.UNION of 2 events A and B, denoted by UNION of 2 events A and B, denoted by
AA⋃⋃B, is the set containing all elements that B, is the set containing all elements that
belong to A or to B or both.belong to A or to B or both.
Example:Example:
E = {1, 2, 3, 4, 5}E = {1, 2, 3, 4, 5}
F = {2, 5, 6, 7, 8}F = {2, 5, 6, 7, 8}
E E ⋃⋃ F = {1, 2, 3, 4, 5, 6, 7, 8} F = {1, 2, 3, 4, 5, 6, 7, 8}

27. Operations On EventsOperations On Events
3.3.Other OperationsOther Operations..

A A ⋃ ⋃ ΩΩ = = ΩΩ

A A ⋂ A' = ⋂ A' = ϕϕ

ΩΩ' = ' = ϕϕ

(A')' = A(A')' = A

A A ⋃ ⋃ ϕϕ = A = A

A A ⋃ A'⋃ A' = = ΩΩ

ϕϕ'' = = ΩΩ