Sample Space in Probability

In Probability Theory, the sample space is the set of all possible outcomes of a random experiment. An experiment is any process that gives a result, like tossing a coin or rolling a die. Each outcome in the sample space can have a probability. A group of one or more outcomes from the sample space is called an event.

Another example of a Sample space for some random experiment:

• Experiment: Spinning a European roulette wheel
• Sample space: {0, 1, 2, 3, 4, 5, ..., 36}

Types of Sample Spaces

There are three types of Sample space:

Finite: The number of outcomes is countable and limited (e.g., rolling a die).
Sample Space: S = {1, 2, 3, 4, 5, 6}

Infinite: The outcomes go on forever (e.g., the Number of coin tosses needed until the first head appears).
Sample Space: S = {1, 2, 3, 4, 5, 6, ... }

Continuous: The outcomes can take any real value within a given range (e.g., A real number between 0 and 1).
Sample space: S = {x ∈ R ∣ 0 ≤ x ≤ 1}

Finding Sample Space in Probability

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

1. Identify all possible outcomes of the experiment.
2. List these outcomes in a set, ensuring each one is unique.

Combining sample spaces when multiple events occur helps calculate complex probabilities.

Sample Space- Tossing Coins

The sample space for tossing coins is given below:

When Two Coins are Tossed

Each coin can result in two possible events: either a head or a tail. In the case of flipping two coins, there are 4 sample spaces given as

(HH), (HT), (TH), (TT)

When Three Coins are Tossed

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 contain 2³ = 8 different outcomes, each with varying numbers of heads and tails in different orders, given as

Sample Space- Rolling Dice

Sample space for rolling a die and two dices are given below:

When One Die is Rolled

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}.

When Two Dice are Rolled

When rolling two dice, the sample space represents all the combinations of outcomes that can occur. It consists of pairs of numbers ranging from (1, 1) to (6, 6), consisting of 6⁶= 36 pairs. It helps in calculating probabilities for various sums or events involving two dice.

The following is a table of the sample space of rolling two dice.

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 dice 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?

Answers:

1. 4
2. 36
3. 270,725
4. 2,704
5. 1,000

Related Articles

• Events in Probability
• Types of Events in Probability
• Bayes Theorem in Probability
• Random Experiment - Probability
• Rolling a Die - Probability, Sample Space, Examples