What is an Experiment?
An experiment can be defined as a procedure with different results. The possible results of any experiment are known as outcomes.
What is a Sample Space?
A sample space can be defined as a set of all possible outcomes of a random experiment. The sample space is represented by the symbol, ‘S’. Sample space can also be expressed using the set notation, {}.
Depending on the experiment, a sample space may have a specific number of outcomes. If the outcomes are finite in number, the sample space is known as a discrete or finite sample space.
What is the difference between Sample Space and Events?
Events are the subsets of a sample space. We can express events as possible outcomes of an experiment as well. Let us take an example of rolling a dice. Here, the possible outcomes will be
1, 2, 3, 4, 5 or 6, which we can write as, S = {1, 2, 3, 4, 5, 6}. These possible outcomes are known as a sample space. Now {1, 3, 5} is the event that represents the odd numbers out of the possible outcomes.
Give an example of Sample Space
One of the classic examples of sample space is the tossing of a coin. When a coin is tossed, the possible outcomes of the experiment are either heads or tails. So, the sample space can be written as, S = {Heads, Tails}
What is a Tree Diagram?
A tree diagram is a method to find the possible outcomes of two or more events. Let us take an example of rolling a dice and tossing a coin. As we know, when we flip a coin, the two possible outcomes are {heads, tails} and six possible outcomes for rolling a dice are {1, 2, 3, 4, 5, 6}. This shows that there are two events: an event of {heads, tails} and another event of {1, 2, 3, 4, 5, 6}. If the coin tossing and rolling of dice are done simultaneously, these are the following possible outcomes:
Heads and 1
Heads and 2
Heads and 3
Heads and 4
Heads and 5
Heads and 6
Tails and 1
Tails and 2
Tails and 3
Tails and 4
Tails and 5
Tails and 6
The sample space can be written as, {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.
Now, these possible outcomes can be represented in the form of a tree diagram. Here, every branch shows a way of acquiring the outcome in a proper way.
Solved Sample Space Examples
Example 1. Margaret is playing a game with her friends where she has to roll two dice at the same time. Find the sample space.
Answer:
When Margaret rolls two dice at the same time, there will be 36 pairs of possible outcomes. In order to determine the sample space, we will find the possible outcomes from this experiment. The 36 outcome pairs are written as follows:
Each face of the first dice can be tossed with the six other faces of the second dice. As there are 6 x 6 possible pairs, there are 36 outcomes.
Example 2: Find the sample space for the given interval: [2, 8]
Answer:
Here, the integers are given in a closed interval, the values taken for consideration are from 2 to 8. Thus, the sample space for the given set of integers is,
S = {2, 3, 4, 5, 6, 7, 8}
Example 3. What is the sample space for choosing from a standard deck of cards?
Answer:
A standard deck of cards will have 4 suits of cards – hearts, clubs, spades, and diamonds.
Now, the sample space for choosing a card from a standard deck of cards is,
{Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.
Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.
Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.
Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A.}
Where, j = joker, q = queen, k = king, and A = ace
A sample space can be defined as a set of possible outcomes for an experiment.
Mutually exclusive events are those events that cannot occur simultaneously. There are no common outcomes among the events.
Equally likely events are those events in which the outcomes are equally likely to occur. Tossing a coin and getting either a heads or tails is a popular example.
A large sample space is when two or more events are executed and a sample space is created for those events.