Chapter 1.1 Random Experiment, Sample Space

Question 1

It is a process that can be repeated under similar conditions but whose outcome cannot be predicted with certainty beforehand

Answer 1

Random Experiment

Question 2

Give examples of a random experiment (repeated under similar conditions but not the same outcome)

Answer 2

Tossing a pair of dice, tossing a coin, selecting cards (It’s not possible to determine in advance what the next outcome will be)

Question 3

TRUE OR FALSE

Can we predict the outcome of the random experiment?

Answer 3

We can’t. Outcomes from random experiments are expected after the experiment has been done already therefore we cannot predict. HOWEVER, we can specify the collection of all its possible outcomes.

For more clarifications, it’s like tossing a coin, when you toss a coin you’re not certain about what side the coin will land on but you know that it’s either a head or a tail (H or T is the specifid collection of the possible outcomes of tossing coin)

Question 4

It is the collection of all possible outcomes of a random experiment

Answer 4

Sample space

Question 5

What is the notation of the sample space?

Answer 5

Ω (Greek letter, omega)

Question 6

The sample space is also known as ——- in set terms. Why is that?

Answer 6

Universal set, because it contains ALL the possible outcomes of a random experiment

Question 7

What are the ways that we can specify a set?

Describing all the possible outcomes in the random experiment

Answer 7

Roster Method and Rule Method

Question 8

Specifying a set:

This lists down all the elements belonging in the set then enclosing them in braces

Answer 8

Roster Method

Question 9

Specifying a set:

States a rule that the elements must satisfy in order to belong in the set then enclosing this rule in braces

Answer 9

Rule Method

Question 10

It is an element of the sample space

(What the sample space contains)

Answer 10

Sample point

Question 11

Roster method

In the sample space: Ω = {HH, TH, HT, TT}
What are the sample points?

Answer 11

HH, TH, HT, TT

It contains 4 sample points that represents the possible outcomes of the random experiment

Question 12

Rule method

How would you describe the sample space of: Ω = {HH, TH, HT, TT}
In the rule method?

Answer 12

Ω = {(x, y) | x ∈ (H,T) and y ∈ (H,T)}

Read as, the sample space with the ordered pair (x, y) such that x is an element of H and T and y is an element of H and T.

Question 13

TRUE OR FALSE

Is the sample space unique?

Answer 13

FALSE, the sample space is not unique because there are many ways we can specify the collection of all possible outcomes of the experiment

Example, when you throw a die, you can describe its outcomes by Ω = (A1, A2, … A6) or you can also define it as Ω = (E, O) even numbers, or odd numbers. They both satisfy the definitions of tossing a die but the way its defined are different.

Question 14

Suppose a population contains 10 distinct elements labeled from 1-10. You conduct a random experiments that selects a sample consisting of 5 elements using SRSWR. What is the sample space?

Answer 14

The sample space is all the possible ways of selecting 5 elements from the population. We can define it as:
Ω = {(X1, X2, …, X5) | Xi ∈ (1, 2, … 10) for all i}

This is just 10x10x10x10x10 (10^5) since all 10 elements have the same equal chances of appearing in all of the 5 that will taken.

Question 15

How do you choose between rule and roster method?

Answer 15

The choice depends on the characteristic of interest and whatever will facilitate the assignment and computation of probabilities

Question 16

What is simple random sampling without replacement (SRSWOR)?

Answer 16

it is all the possible subsets consisting of n distinct elements from the N elements of the population that have the same chances of selection

Question 17

It is a SRS whose chances have distinct elements

Answer 17

SRSWOR (simple random sampling without replacement)

• Contains n distinct elements.
• Order does not matter ({a,c} is the same as {c,a})
• Uses combination

Question 18

What is simple random sampling with
replacement (SRSWR)?

Answer 18

contains all the possible ordered n-tuples (coordinates need not be distinct) that can be formed from the N elements of the population have the same chances of selection.

Question 19

It is a SRS whose chances have ordered n-tuples

Answer 19

SRSWR (simple random sampling with replacement)

• Contains ordered n-tuples
• Order matters ({a,c} is not the same as {c,a})
• Uses permutation with repetition

Question 20

TRUE OR FALSE

In both SRSWR and SRSWOR do the sample points/elements of the sample have the same chances of selection

Answer 20

Yes, TRUE

Question 21

It is a subset of the sample space whose probability is defined

Answer 21

Event

Question 22

What is an event?

Answer 22

It is a subset of the sample space whose probability is defined

Question 23

When can you say that an event occurred? if the outcome of the experiment is one of the sample points belonging in the event

Otherwise, it did not occurr

Answer 23

If the outcome of the experiment is one of the sample points belonging in the event

We denote the events by using the Latin capital letter (A, B, C….)

Question 24

What are the two subsets of the sample space that will always be events?

Answer 24

the empty set and the sample space itself

Question 25

What is the **impossible event**

Answer 25

It is also referred to as the empty set

Question 26

It is also referred to as the **sure event** in sets

Answer 26

sample space

Question 27

# Other events The collection of sample points in the sample space that do not belong in A ## Footnote Occured if event A did not occur

Answer 27

A' or A^c (read as **A complement**)

Question 28

# Other events The collection of sample points that belong in at least one of A and B ## Footnote occured if only event A occured, only event B, or both A and B

Answer 28

(A U B), read as "A union B" ## Footnote (A U B) is also the same as "A or B"

Question 29

# Other events Collection of sample points that belong in both A and B ## Footnote occured if both events A and B occured simultaneously

Answer 29

(A ^ B), read as "A intersection B"

Question 30

# Other events collection of sample points that belong in at least one of A1, A2, ... An ## Footnote Occured if at least one of the n events occured

Answer 30

(A1 U A2 .... UAn), read as "the union of n events"

Question 31

# Other events the collection of sample points that belong in each one of A1, A2, ..., An

Answer 31

(A1 ^ A2, ... ^A2), read as "the intersection of n events"

Question 32

What do you call the relationship of two events that have no elements in common?

Answer 32

Mutually exclusive events, (A ^ B) = ∅

Question 33

# TRUE OR FALSE Any collection is **mutually exclusive** if the collection is a **pairwise disjoint**

Answer 33

TRUE

Question 34

What is a pairwise disjoint?

Answer 34

it is when **one event in the collection occurs** **then any other of the other events in the collection canot occur.**