A sample space is a collection or a set of possible outcomes of a random experiment. The sample space is represented using the symbol, “S”. The subset of possible outcomes of an experiment is called events. A sample space may contain a number of outcomes which depends on the experiment. If it contains a finite number of outcomes, then it is known as discrete or finite sample spaces.
The samples spaces for a random experiment is written within curly braces “ { } “. There is a difference between the sample space and the events. For rolling a die, we will get the sample space, S as {1, 2, 3, 4, 5, 6 } whereas the event can be written as {1, 3, 5 } which represents the set of odd numbers and { 2, 4, 6 } which represents the set of even numbers. The outcomes of an experiment are random and the sample space becomes the universal set for some particular experiments. Some of the examples are as follows:
Tossing a Coin
When flipping a coin, two outcomes are possible, such as head and tail. Therefore the sample space for this experiment is given as
Sample Space,S = { H, T } = { Head, Tail }
Tossing Two Coins
When flipping two coins, the number of possible outcomes are four. Let, H_{1} and T_{1 }be the head and tail of the first coin and H_{2} and T_{2} be the head and tail of the second coin respectively and the sample space can be written as
Sample Space, S = { (H_{1}, H_{2}), (H_{1}, T_{2}), (T_{1}, H_{2}), (T_{1}, T_{2}) }
In general, if you have “n” coins, then the possible number of outcomes will be 2^{n}.
Example: If you toss 3 coins, “n” is taken as 3.
Therefore, the possible number of outcomes will be 2^{3} = 8 outcomes
Sample space for tossing three coins is written as
Sample space S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
A Die is Thrown
When a single die is thrown, it has 6 outcomes since it has 6 faces. Therefore, the sample is given as
S = { 1, 2, 3, 4, 5, 6}
Two Dice are Thrown
When two dice are thrown together, we will get 36 pairs of possible outcomes. Each face of the first die can fall with all the six faces of the second die. As there are 6 x 6 possible pairs, it becomes 36 outcomes. The 36 outcome pairs are written as:
\(\begin{Bmatrix} (1,1)&(1,2)&(1,3)&(1,4)&(1,5)&(1,6) \\ (2,1)&(2,2)&(2,3)&(2,4)&(2,5)&(2,6) \\ (3,1)&(3,2)&(3,3)&(3,4)&(3,5)&(3,6)\\ (4,1)&(4,2)&(4,3)&(4,4)&(4,5)&(4,6) \\ (5,1)&(5,2)&(5,3)&(5,4)&(5,5)&(5,6) \\ (6,1)&(6,2)&(6,3)&(6,4)&(6,5)&(6,6) \end{Bmatrix}\)If three dice are thrown, it should have the possible outcomes of 216 where n in the experiment is taken as 3, so it becomes 6^{3} = 216.
Sample Problem
Question:
Write the sample space for the given interval [3,9]
Solution:
Given interval: [3, 9]
As the integers given are in the closed interval, we can take the value from 3 to 9
Therefore, the sample space for the given intervals are:
Sample space = { 3, 4, 5, 6, 7, 8, 9 }
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