1101 LearnersLast updated on June 18th, 2025
Sample Space
Sample space is a set of all possible outcomes of an experiment. We use sample space in real-life uncertain situations such as drawing a particular card from a deck or guessing the correct answer in multiple-choice questions. In this topic, we will learn about sample space from a broad perspective.
What is Sample Space in Probability
The sample space is a mathematical concept that refers to the possibility of the occurrences of different outcomes in a random experiment. Sample space is often represented by “S”.
For example, the sample space for rolling a die is S = {1, 2, 3, 4, 5, 6}. Understanding random events is essential for calculating probabilities correctly.
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How to Find Sample Space in Probability
The probability of the sample place can be determined using the following steps:
- Identify the purpose of the experiment. For example: Rolling a die.
- List the possible outcomes of the given experiment in a set, ensuring no repetition.
- For two coin flips, the sample space: S = {HH, HT, TH, TT}
- When dealing with complex probabilities, you can combine sample spaces by including all possible outcomes to simplify calculations.
Difference Between Event and Sample Space
Both event and sample space are significant elements of an experiment. The key differences between them are discussed below:
| Sample Space | Event |
| Collection of all possible outcomes of a given experiment. | Collection of specific outcomes from an experiment. |
| Denoted by ‘S’ | Denoted by ‘E’ or other symbols. |
| Includes all possible outcomes | Concentrates on specific outcomes. |
| Applied to determine the probability of total possible outcomes. | Used to determine the probability of a specific outcome. |
| For example: The sample space for a single coin flip: {H, T} |
For example: Obtaining tails: E = {T} |
Real-life applications
Sample space is a significant concept that lists all the possible options for an event. It can be applied in various fields to determine probability. Now, let’s explore the real-life applications of Sample Space:
- Sample space is widely used in sports to decide which team should choose batting or fielding, where the sample space is{Heads, Tails}.
- Traffic light signals showcase all possible outcomes in a traffic light system. {Red, Yellow, Green}.
- In the medical field, the sample space is used in diagnosing a disease, S= {Positive, Negative}.
- Children use sample space in rolling a die in board games like Ludo or Snakes and Ladders. The sample space is {1,2, 3, 4, 5,6}.
- Students can make decisions, such as choosing which subject needs to be studied first based on the sample space{ Math, Science, Social}.
Common Mistakes and How to Avoid Them in Sample Space
Sample space is an important concept that helps students predict the probability of uncertain events. However, some students make mistakes when working with problems related to sample spaces. Here, we list some common mistakes along with ways to master the concept easily:
Mistake 1
Missing Possible Outcomes
Some students mistakenly fail to include all possible outcomes.
For example: If the experiment is rolling a die, they might write the sample space as S = {1, 2, 3, 4, 5}(Where 6 is missing).
Always check that all possible outcomes listed are correct before confirming the sample set.
Mistake 2
Incorrect Order in Events
Overlooking the order of events can lead to inaccurate sample space.
For example, The sample space for blue and red pens in a box: S = {BB, RR}but excluding
{BR, RB}.
Ensure that you follow a particular order that helps you spot if a pair is missing.
Mistake 3
Confusion between Sample Space and Probability
Some students mistakenly assume that probability is the same as the sample space.
For example, They incorrectly write S = 1/6 instead of writing all the possible results.
Keep in mind that sample space lists all the possible outcomes, whereas probability determines the chance of getting a specific outcome.
Mistake 4
Repetition of Outcomes
Writing the same outcomes repeatedly in a set results in an inaccurate sample space.
For example, for a single coin flip, some students write: {H, H, T, T} instead of {H, T}.
Note that, the sample space always contains unique terms or numbers.
Mistake 5
Incorporating Impossible Outcomes
In some cases, students include impossible outcomes in the sample space.
To avoid confusion, identify what needs to be determined in the experiment.
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Solved Examples of Sample Space
Problem 1
Lena has two shirts (Black, Red) and two pairs of skirts (Blue, White). Determine the sample space for their outfit choices.
S = { (Black, Blue), (Black, White), (Red, Blue), (Red, White)}
Explanation
The first step is to identify the experiment:
Choosing a combination of one shirt and one pair of skirts.
Here, we list all the possible combinations:
- Black shirt with Blue skirt
- Black shirt with White skirt
- Red shirt with Blue skirt
- Red shirt with White skirt
Now, we get the sample space as:
S = { (Black, Blue), (Black, White), (Red, Blue), (Red, White)}.
Problem 2
John arrives at a bus stop where three subways (Subway 1, Subway 2, Subway 3, and Subway 4) can arrive next. What is the sample space?
The sample space we get is: {Subway 1, Subway 2, Subway 3, Subway 4}
Explanation
The experiment needs to determine which subway arrives next.
Here, we list all possible outcomes:
- Subway 1
- Subway 2
- Subway 3
- Subway 4
Now, the sample space we get is: {Subway 1, Subway 2, Subway 3, Subway 4}.
Problem 3
A teacher asks students to pick a paper lot numbered from 1 to 6. What is the sample space?
S = {1, 2, 3, 4, 5, 6}
Explanation
Here, the experiment is students picking a paper lot numbered from 1 to 6.
We will now list all possible numbers written on the paper lots:
- 1
- 2
- 3
- 4
- 5
- 6
Thus, the sample space is S = {1, 2, 3, 4, 5, 6}.
Problem 4
A café offers four varieties of ice cream flavors such as corn, mint chocolate chip, bubblegum, and cinnamon apple. If a customer randomly picks a random flavor of ice cream, what is the sample space?
S ={Corn, Mint chocolate chip, Bubblegum, Cinnamon apple}
Explanation
Here, the experiment is to determine the sample space, where a customer randomly picks an ice cream flavor.
The possible flavors we have:
- Corn
- Mint chocolate chip
- Bubblegum
- Cinnamon apple
Therefore, the sample space is:
S ={Corn, Mint chocolate chip, Bubblegum, Cinnamon apple}
Problem 5
Angel randomly chooses a mode of transportation to go to the office. The available options are walking, a two-wheeler, a cab, and a car. What is the sample space?
S = {Walking, Two-wheeler, Car, Cab}
Explanation
The experiment is to determine the selection of a mode of transportation to go to the office.
All the available options are listed:
- Walking
- Two-wheeler
- Cab
- Car
Thus, the sample space including all possible outcomes is:
S = {Walking, Two-wheeler, Car, Cab}
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FAQs of Sample Space
1.Give one major distinction between sample space and an event.
2.What would be the sample space if a coin is flipped?
3.Is it possible for sample space to be infinite?
4.Can we determine the sample space in real life?
5.What is the significance of sample space in probability?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
