Sample Space refers to the set of all possible outcomes of a random experiment or process. When a die is rolled, the total number of elements in the sample space is 6 while when a coin is tossed, there are a total of two possible outcomes.

Let’s learn how to find the Sample Space of Rolling a Die and Tossing a Coin together and separately, with the help of examples.

Sample Space Definition

The sample space S of a random experiment is the set of all possible outcomes that can occur in a experiment

It is denoted by the symbol S.

Learn, Sample Space in Probability

Sample Space of Rolling a Die

When rolling a fair six-sided die, the sample space (S) is the set of all possible outcomes. A six-sided die has six faces, each numbered from 1 to 6.

Sample space of Rolling a Die can be represented as:

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

Here, each number in the set represents one possible outcome when rolling the die.

Learn, Rolling A Die

Sample Space of Rolling Two Die

The sample space of rolling two fair six-sided dice is obtained by considering all possible combinations of outcomes from the two dice.

To represent the sample space, you can use an ordered pair (a,b), where a is the outcome of the first die a and b is the outcome of the second die.

Since each die has six faces, the sample space of Rolling Two Die is:

S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),

(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),

(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),

(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),

(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),

(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

Sample Space of Tossing a Coin

Since a coin has two distinct sides, heads (H) and tails (T), the sample space for a single coin toss is:

S = {H,T}

Here, each element in the set represents one possible outcome when tossing the coin.

Learn, Coin Toss Probability

Sample Space of Tossing Two Coins

The sample space (S) for rolling two coins can be represented using ordered pairs, where the first element corresponds to the outcome of the first coin, and the second element corresponds to the outcome of the second coin.

Each coin has two possible outcomes.

Sample space of Tossing Two Coins is as follows:

S={(H,H),(H,T),(T,H),(T,T)}

Here, each ordered pair represents a possible combination of outcomes when rolling two coins. There are a total of 2×2=4 possible outcomes in the sample space.

Sample Space of Tossing Three Coins

The sample space (S) for rolling three coins can be represented using combinations of the possible outcomes for each coin. There are two outcomes for each coin, and there are three coins,.

Sample space of Tossing Three Coins is as follows:

S={(H,H,H),(H,H,T),(H,T,H),(T,H,H),(H,T,T),(T,H,T),(T,T,H),(T,T,T)}

Sample Space for Tossing Four Coins

The sample space (S) for rolling four coins can be represented using combinations of the possible outcomes for each coin.

Since there are two outcomes for each coin, and there are four coins, the sample space is as follows:

S = {(H,H,H,H),(H,H,H,T),(H,H,T,H),(H,T,H,H),(T,H,H,H),(H,H,T,T),(H,T,H,T),(T,H,H,T),(H,T,T,H),(T,H,T,H),(T,T,H,H),(H,T,T,T),(T,H,T,T),(T,T,H,T),(T,T,T,H),(T,T,T,T)

Sample Space of Rolling a Die and Tossing a Coin Together

To find the combined sample space S of rolling a die and tossing a coin, we need to consider all possible combinations of outcomes from the two events i.e Rolling a die and Tossing a coin.

Sample Space of Rolling a Die and Tossing a Coin Together will be,

S = { 1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T}

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Sample Space of a Die and a Coin – Solved Examples

Example 1. Find the Sample Space:

Consider rolling a die and tossing a coin. Determine the sample space for the combined events.

Solution:

Let D be the sample space for rolling a die: D={1,2,3,4,5,6}.

Let C be the sample space for tossing a coin: C={H,T}.

The combined sample space (S) is given by the Cartesian product of D and C:

S=D×C={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)

So, the sample space is S={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)}.

Example 2. You roll a fair six-sided die and toss a fair coin. Find the probability of getting a prime number on the die and getting Tails in the coin toss.

Solution:

Let A be the event of getting a prime number on the die, and B be the event of getting Tails in the coin toss.

A={2,3,5}

B={Tails}

P(A and B) : P(A)*P(B)

P(A) = (Number of outcomes in A/Total number of outcomes on the die) = 3/6 = 1/2

P(B) = 1/2……………..(As it is a fair coin so)

P(A and B) = (1/2)*(1/2) = 1/4

Therefore: P(A and B) = 1/4

Example 3. You roll a fair six-sided die and toss a fair coin. What is the probability of getting Heads or getting an even number?

Solution:

Let C be the event of rolling an even number, and D be the event of getting Heads.

C = {2,4,6}

D = {Heads}

The probability of either event occurring is given by:

P(C or D) = P(C)+P(D) – P(C and D)

P(C) = 3/6 = 1/2

P(D) = 1/2

P(C and D) = 1/6(the outcomr(2,heads))

P(C or D) = 1/2+1/2-1/6 = 5/6

Therefore, the probability of rolling an even number or getting Heads is P(C or D) = 5/6

Sample Space of a Die and a Coin – Practice Problems

1. You roll a fair six-sided die and flip a fair coin twice. Determine the sample space for this event.

2. You draw two cards without replacement from a standard deck of 52 playing cards. Determine the sample space for this event.

3. You roll two fair six-sided dice. Determine the sample space for this event

Sample Space of a Die and a Coin – FAQs

What is Sample Space?

The sample space (S) is the set of all possible outcomes of an experiment.

What is the Sample Space for Rolling a Die and Tossing a Coin simultaneously?

The sample space for rolling a die and tossing a coin together consists of 12 outcomes: {(1, H), (1, T), (2, H), (2, T), (3, H), (3, T), (4, H), (4, T), (5, H), (5, T), (6, H), (6, T)}, where the first element in each pair is the die roll (1-6) and the second is the coin toss (H for heads, T for tails).

How is Sample Space determined?

The sample space is determined by listing all possible outcomes of an experiment or by using the Cartesian product of individual outcomes for each component of the experiment.

What is difference between an Event and a Sample Point?

A sample point is a single outcome of an experiment, while an event is a subset of the sample space, consisting of one or more sample points.

How is Probability Calculated from the Sample Space?

Probability (P) is calculated by dividing the number of favorable outcomes by the total number of possible outcomes: P(A) = (Number of favorable outcomes)/(Total number of possible outcomes)