# What is sample space: Definition and 39 Discussions

In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.A subset of the sample space is an event, denoted by

E

{\displaystyle E}
. If the outcome of an experiment is included in

E

{\displaystyle E}
, then event

E

{\displaystyle E}
has occurred.For example, if the experiment is tossing a single coin, the sample space is the set

{
H
,
T
}

{\displaystyle \{H,T\}}
, where the outcome

H

{\displaystyle H}
means that the coin is heads and the outcome

T

{\displaystyle T}
means that the coin is tails. The possible events are

E
=
{
H
}

{\displaystyle E=\{H\}}
,

E
=
{
T
}

{\displaystyle E=\{T\}}
, and

E
=
{
H
,
T
}

{\displaystyle E=\{H,T\}}
. For tossing two coins, the sample space is

{
H
H
,
H
T
,
T
H
,
T
T
}

{\displaystyle \{HH,HT,TH,TT\}}
, where the outcome is

H
H

{\displaystyle HH}

H
T

{\displaystyle HT}
if the first coin is heads and the second is tails,

T
H

{\displaystyle TH}
if the first coin is tails and the second is heads, and

T
T

{\displaystyle TT}
if both coins are tails. The event that at least one of the coins is heads is given by

E
=
{
H
H
,
H
T
,
T
H
}

{\displaystyle E=\{HH,HT,TH\}}
.
For tossing a single six-sided die one time, where the result of interest is the number of pips facing up, the sample space is

{
1
,
2
,
3
,
4
,
5
,
6
}

{\displaystyle \{1,2,3,4,5,6\}}
.A well-defined, non-empty sample space

S

{\displaystyle S}
is one of three components in a probabilistic model (a probability space). The other two basic elements are: a well-defined set of possible events (an event space), which is typically the power set of

S

{\displaystyle S}
if

S

{\displaystyle S}
is discrete or a σ-algebra on

S

{\displaystyle S}
if it is continuous, and a probability assigned to each event (a probability measure function).

A sample space can be represented visually by a rectangle, with the outcomes of the sample space denoted by points within the rectangle. The events may be represented by ovals, where the points enclosed within the oval make up the event.

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1. ### I Random variable and elementary events

Hello, A sample space is the set of all possible elementary events. A random "variable" is really a real-valued function that associates a single real number to every elementary events. For example, in the case of a fair die, the sample space is ##\Omega={1,2,3,4,5,6}##. Each number is an...
2. ### I Sample space, outcome, event, random variable, probability...

Hello, I am solid on the following concepts but less certain on the correct understanding of what a random variable is... Random Experiment: an experiment that has an uncertain outcome. Trials: how many times we sequentially repeat a random experiment. Sample space ##S##: the set of ALL...
3. ### Are both sample spaces the same or do they mean different things?

I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up. $$S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \}$$  S = \{ (i,j) : i...
4. ### MHB Sample space and bridge game

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5. ### A Probability and entropy in an exponentially increasing sample space

Hi, I'm new to PF and not really sure which forum may be the most appropriate to find people with an interest in probability and entropy. But the title of this forum looks promising. If you share an interest in this topic would be delighted to hear from you.
6. ### Equiprobable sample space: N Identical objects in K different containers

So we had an exam question which was the following: Assume you have N identical balls and K different contains if there are N pre-Selected boxes and that N < k what is the probablity that none of these pre-selected boxes are empty? I answered it and it was the same as the professor's answer...
7. ### MHB Probabilities involving dynamic sample space

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8. ### What are the probabilities for events A and B in a set of 10 cards?

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9. ### Probability and Stats - Show the Sample Space

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10. ### Sample space probability question

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11. ### MHB Draw the sample space & find the probabilty

Having trouble in drawing the sample space in the grid :D
12. ### Probability math confusion sample space (I think)

Homework Statement high school has 417 students total 186 of total are athletes (play sports) 136 of total are musicians (play music) 74 of total are musicians and athletes. (play music and play sports) a) at which probability does randomly chosen athlete also play music (i.e. be a musician)...
13. ### Show that a sample space is valid by verifying properties

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14. ### Random experiment and sample space

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15. ### Multivariable sample space

Hi, I'm currently having a lot of trouble with this probability problem. For example: Suppose there are 5 balls in a bag with number 1,2,3,4,5. I pick a ball at random 20 times (with replacement). Lets say the probability of each ball being picked is: P(1) = 0.5 P(2) = 0.15 P(3) = 0.1 P(4) = 0.2...
16. ### Is the sample space not a set under ZFL?

I am reading Introduction to Set Theory (Jech & Hrbacek) and in one of the exercises we're asked to prove that the complement of a set is not a set. I get that if it were a set that would imply that "a set of all sets" (the union of the set and its complement, by the axiom of pairing) exists and...
17. ### MHB Help with Probability & Sample Space Questions

I have been having a time trying to get the answers for these two questions. Can anyone please help me? 1) Suppose a fair die is tossed and the number showing on the top face is recorded. Let E, F, and G be the following events: E: {1,2,3,5}, F:{2,4}, G:{1,4,6} Compute the probability of the...
18. ### Sigma Algebra on Omega (Sample Space)

Respected Members, Suppose Ω is the set of eight possible outcomes of three coin tosses i.e. Ω={{HHH, HHT, HTH, HTTT, THH, THT, TTH, TTT} So if we are not told the results then the sigma algebra ( denoted by F_α) at position α=0 is F_0 = {∅, Ω} Now if are told the first coin toss only...
19. ### MHB -aux.02 Venn diagram sample space U and events A and B

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20. ### Probability - Independent events with minimal points in sample space

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21. ### Is Sample Space an Open or Closed Set?

Hi all, is sample space an open set or a closed set? Thanks in advance
22. ### Elements of sample space

Sample space of rolling a 6 sided die I've just had my first tutorial for an introductory probability class taught by the Econometrics department and I'm having trouble understanding why my solution is wrong. The tutor didn't write up complete solutions(just pictures to help visualise the...
23. ### Random Process vs Random Variable vs Sample Space

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24. ### Cardinality of Sample Space for Brownian Motion

Hello, Given a Brownian Motion process B(t) for 0≤t≤T, we can write it more explicitly as B(t,ω) where ω\inΩ, where Ω is the underlying sample space. My question is: what is the cardinality of Ω. I.e. what is |Ω|? My thoughts are that it is an uncountable set, based on the observation...
25. ### Stats - find the distribution function of an infinite sample space.

Homework Statement A die is rolled until the first time that a six turns up. We shall see that the probability that this occurs on the nth roll is (5/6)n−1 · (1/6). Using this fact, describe the appropriate infinite sample space and distribution function for the experiment of rolling a die...

I am a beginner to quantum mechanics and am trying to make sense of Schrodinger's Equation. I am attempting to find probabilities in the case of a free particle in the general case. It is my understanding that the solution to Schrodinger's Equation in the general case of a free particle is as...
27. ### Understanding Sample Space: Explaining a Tricky Dice Question

A six sided dice has the numbers 1,1,1,2,3,4. What is the sample space? Is it 1,2,3,4 or 1,1,1,2,3,4? Could you please explain why? All of my classmates are arguing about this question.
28. ### Outcome Space and Sample Space

Hi, I am new to this forum. I was wondering if someone could explain to me what is the difference between the outcome space and the sample space. My teacher gave me the definition but I can't seem to understand what is the difference. He also gave me an example but I need help with it too...
29. ### Sample space and probability measure.

If we have two red balls and we are to choose one it is true that the probability measure of picking a red ball is 1. In this case it is understood that the sample space only contains a red ball and this is because events in the sample space are to be disjoint. But how come this is not the...
30. ### Difference between Power Sets and Sample Space

Hey All, In my probability theory class we have just started learning about how a probability space is defined by a sample space (which contains all possible events), events and a measure. We briefly went over the idea of the Power Set, which seems to be the set of all subsets in your...
31. ### Two dimensional sample space

Let’s select two numbers x and y such that 1<=x<=4 and 2<=y<=6. What is the probability of x +y>=5? I solved the problem and attached it. Please check my solution and tell me whether my approach of solution is correct or wrong. The most puzzling matter is that I'm getting two different...
32. ### Probability - Infinite Union of Subsets of a Sample Space

Homework Statement This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach. Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t} where Y is a random variable and t is any real number. Fix a real number...
33. ### Possibility of a sample space containing the following:

[SOLVED] Possibility of a sample space containing the following: Will it ever be possible for a sample space (the set of all possible outcomes of a probability experiment) to contain outcomes that are both independent of each other but in fact also mutually exclusive? Please note I am...
34. ### Amplitude for paths in sample space?

Is there any study anywhere considering paths through a sample space from one element to another? Is there an "amplitude" associated with these paths?
35. P

### What is the sample space for randomly placing 8 pieces on an 8x8 checkerboard?

The question asks: Suppose that we randomly place 8 playing spieces on an 8 x 8 checkerboard. What is the probability that each row of the checkerboard will have only one piece? I've been trying forever to get this question going. My problem is that I cannot figure out the magnitude of...
36. ### Sample Space for Random Exp: {1,2,3,4,5,6}

Determine the sample space for this random experiment: An urn contains six balls numbered 1-6. The random experiment consists of selecting five balls without replacement.
37. ### Sample Space of Selecting 5 Balls Without Replacement

Determine the sample space for this random experiment: An urn contains six balls numbered 1-6. The random experiment consists of selecting five balls without replacement. The way i did it was by figuring that the balls were selected simultaneously so i got six different combinations. But...
38. ### Sample Space for Conventional Knock-Out Tournament

Somebody could help me with this question? A conventional knock-out tournament begins with 2^n competitors and has n rounds. There are no play-offs for the positions 2, 3, ..., 2^(n)-1, and the initial table of draws is specified. Give a concise description of the sample space of all possible...
39. ### Neep help in the sample space:

1. Suppose that two cards are dealt from a standard 52-card poker deck. Let A be the event that the sum of the two cards is 8 (assume that aces have a numerical value of 1). How many outcomes are in A? Where I got stuck [WIGS]: Are the suits important here? So, there are a lot of...