Sample Space

What is Sample Space? 

If you want to understand probability in-depth, you will have to start with the concept of sample space. In easy words, sample space is said to be the total number of things on which the experiment has to be done. The definition of sample space can be different in different scenarios as per the requirement of the question or the problem. 

Probability can be applied to a variety of problems and scenarios. Now, let us discuss each case and scenario in detail to understand the role of sample space and how it varies from case to case.

The first case involves playing cards. We know for a fact that the total number of playing cards in a deck is fifty-two. So the sample space in the case of playing cards is fifty-two. Some more examples in this case regarding the sample space are- : 

If we are asked to find the probability of getting any face card in a deck then it will be twelve upon fifty-two. Now there are twelve face cards in a deck including four kings, queens, and jack. Fifty-two came in the denominator as the sample space or the total number of cards is fifty-two. 

More examples regarding sample space 

The second example regarding sample space can be considered in the game of pool that includes two types of coloured balls named Stripes and Solids. The stripes are the balls having different colour combinations and the solids are the ones having only a single colour. The total number of balls on the pool table is fifteen out of which, if we exclude the black ball, we have seven striped balls. 

Concept of number of balls in a table

So the probability of getting a striped ball is seven upon fifteen. Again, to explain how we reached this expression, seven is the number of striped balls and fifteen is the number of total balls or the sample space. Hence, from the above two examples, it’s clear that the sample space is always in the denominator. 

Another very famous example of probability is that of numbers. If we take the sample space to be numbers from one to hundred, then the probability of getting an odd or an even number is exactly half as fifty of the hundred numbers are either odd or even. So independent of the outcome, the probability will come out to be the same. If we want to take out the probability that a number will be divisible by three, then the probability will come out to be thirty-three upon hundred as hundred is the sample space and thirty-three is the number of numbers that are divisible out of hundred. 

Understanding Sample Space in depth 

After all the above examples of sample spaces in different scenarios, let us now express sample space with the help of sets. The most common example in the case of sample space is that of a coin toss. The set of a sample space is always denoted by E. If a single coin is tossed, then there are only two outcomes H which denotes head and T which denotes tails. So, in the case of a single coin toss, E = {H, T}. 

Concept of a two-coin and a three-coin toss

The aforesaid expression states that the sample space of a single coin toss includes a single outcome for each of heads and tails. Now, if we take the example of a two-coin toss, the sample space will turn out to be more complicated than the one coin toss. The total number of outcomes in the case of two coin tosses will be four including HH, TT, HT, and TH. 

Hence, the outcomes can be presented in the form, E={ HH, TT, HT, TH}. Now to conclude this set theory, let us take the example of a three-coin toss. Now the number of outcomes, in this case, will be eight, so it can be expressed in the following way also.

E={HHH,TTT,HHT,HTT,HTH,TTH,THT,THH}. 

A very easy way to figure out the sample space is to simply put the number of coins on the power to two. Like, in the case of three coin toss, if we see that two to the power three comes out to be eight which was exactly the number of outcomes in sample space, and in the case of a two-coin toss, two to the power two which comes out to be four, was also the number of outcomes of in the sample space. 

Concept of Multiple Sample Spaces 

If we are to understand multiple sample spaces, then we will have to consider events that have the possibility of multiple sample spaces like the card games wherein we can have one set of normal cards and one set of face cards. If we want to have an outcome of the face cards like to figure out the probability of the face cards in a deck, then we can refer to the latter set and if we want to have a problem with the earlier set then we can go for the first set. 

Conclusion 

Sample space is one of the most important concepts when we get into probability and try to study it in depth. Without sample space, the probability will not have any sense as we will not have any sample data.