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 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.