Sample Space in Probability- Definition and Solved Examples

Sample Space in Probability- Sample Space is a set of all possible outcomes of a random experiment. The subset of possible outcomes of an experiment is called events. In this article, we will discuss what is Sample Space in probability, its meaning, examples and definition, how to find sample space in probability, and sample space for rolling a die and two dice, along with some solved examples and practice problems on sample space in Probability.

What is Sample Space in Probability

Sample Space is a concept in probability theory that deals with the likelihood of different outcomes occurring in a given experiment. It involves defining a sample space that encompasses all possible outcomes and assigning probabilities to these outcomes.

For example, when rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. In a coin toss, the sample space is {Heads, Tails}. Sample space is crucial for calculating probabilities and understanding random events.

Sample Space is a fundamental concept in Probability Theory.

Sample Space Math Definition

In mathematics, the sample space is a set that contains all possible outcomes of a random experiment or event.

Sample Space is a key concept in probability theory and is used to determine the likelihood of different results occurring in a random experiment or event, by representing all possible outcomes or events that can occur.

Sample Space Meaning

The sample space in probability refers to the set of all possible outcomes or results that can arise from a random experiment. It serves as the foundation for calculating probabilities and understanding the variability of outcomes.

Example of Sample Space

Some examples of sample space are:

• A classic example of a sample space is a deck of playing cards. The sample space for drawing a single card is {Ace of Diamond, 2 of Hearts, King of Spades}.
• When tossing two coins, the sample space is {(H, H), (H, T), (T, H), (T, T)}.
• Rolling two dice results in a sample space of {(1, 1), (1, 2),(1, 3), (1, 4), . . . (6, 6)}.

Sample spaces vary depending on the experiment and help analyse possible outcomes.

How to Find Sample Space in Probability

To find the sample space in Probability, follow the below steps:

1. Identify all possible outcomes of the experiment.
2. List these outcomes in a set, ensuring each one is unique.
3. For a single die roll, the sample space is {1, 2, 3, 4, 5, 6}.
4. For drawing a card from a standard deck, the sample space is 52 unique cards.
5. Combining sample spaces when multiple events occur helps calculate complex probabilities.

Sample Space for 2 Coins

• Each coin can result in two possible events either head or tail.
• In case of flipping two coins there are 4 sample space given as (HH), (HT), (TH), (TT)

Sample Space for 3 Coins

Sample Space for Rolling 3 coins can be calculated keeping in mind the following:

• When flipping three coins, the sample space encompasses all the possible combinations of heads and tails for the three coins.
• It can be containing 2³ = 8 different outcomes each with varying numbers of heads and tails.

Sample Space for Rolling A Die

On rolling a die, we can have 6 outcomes. So the sample space for rolling a die will be, S = {1, 2, 3, 4, 5, 6}.

Sample Space for Two Dice

Sample Space for Rolling Two Dice is as follows:

• When rolling two dice, the sample space represents all the combinations of outcomes that can occur.
• Sample Space for Rolling Two Dice consists of pairs of numbers ranging from (1,1) to (6,6) and helps in calculating probabilities for various sums or events involving two dice.

What is Sample Space Diagram

A sample space diagram is a visual representation that illustrates all the possible outcomes of a random experiment. It is a valuable tool in probability theory for visualising and understanding the different potential results of an event.

Sample Space Diagram for Tossing 3 Coins

Following illustration represents all the possible outcomes i.e., sample space of three coin tossing.

Sample Space Diagram for Rolling of Two Die

Following illustration represents all the possible outcomes i.e., sample space of rolling of two die.

Solved Examples on Sample Space in Probability

Here are some Solved Examples on Sample Space in Probability for you to learn and practise:

Example 1: How many possible outcomes are there when rolling a fair six-sided die?

Solution:

There are 6 possible outcomes when rolling a fair six-sided die.

Example 2: In a deck of 52 playing cards, how many different ways can you draw two cards without replacement?

Solution:

There are 2,652 different ways to draw two cards from a deck of 52 playing cards without replacement.

Example 3: If you flip a coin three times, how many elements are in the sample space for this experiment?

Solution:

There are 2³ = 8 elements in the sample space when flipping a coin three times.

Example 4: A jar contains 20 red marbles and 30 blue marbles. If you draw two marbles without replacement, how many different pairs can you get?

Solution:

There are ²⁰C₁ (choosing 1 red marble) ׳⁰C₁ (choosing 1 blue marble) = 20 × 30 = 600 different pairs you can get when drawing two marbles without replacement.

Example 5: If you have a 4-digit PIN code, and each digit can be 0-9, how many possible PIN combinations are there?

Solution:

There are 10,000 possible PIN combinations for a 4-digit PIN code when each digit can be 0-9.

Practice Problems on Sample Space in Probability

Here are a few Practice Problems on Sample Space in Probability for you to solve:

Problem 1: If you flip a coin two times, how many elements are in the sample space for this experiment?

Problem 2: How many possible outcomes are there when rolling two fair six-sided die simultaneously ?

Problem 3: In a deck of 52 playing cards, how many different ways can you draw four cards without replacement?

Problem 4: In a deck of 52 playing cards, how many different ways can you draw two cards with replacement?

Problem 5: If you have a 3-digit PIN code, and each digit can be 0-9, how many possible PIN combinations are there?

Sample Space in Probability – FAQs

1. What is the definition of a Sample Space?

A sample space is the set of all possible outcomes or results of an experiment or random event.

2. How do you calculate the Size of a Sample Space?

The size of a sample space is determined by counting the number of distinct and equally likely outcomes in a given experiment.

3. What is Sample Space for a Coin Toss?

For a fair coin toss, the sample space consists of two outcomes: heads and tails.

4. Can a Sample Space have Infinite Elements?

Yes, in some cases, a sample space can have an infinite number of possible outcomes such as when dealing with real numbers in a continuous random variable.

5. What is the Sample space in probability cards?

Sample space is fundamental in probability theory as it helps define the likelihood of different events occurring. By understanding the sample space, you can calculate probabilities and make informed decisions in various situations.

6. What is Sample Space Formula?

Sample Space refers to the method of listing or defining all possible outcomes for a given experiment which is important for calculating probabilities. The formula varies depending on the specific problem or experiment.

7. What is an example of space in probability?

If you toss a coin twice, the sample space of this experiment is {HH,HT,TH,TT}.