Sample Space in Probability - Examples and solutions

Sample Space

The sample space plays a very important role in finding the probability of an event associated with a random experiment. Sample space is a term associated with probability theory.
In an experiment or any random trial, when we make a set of all the results or outcomes that are possible in that experiment or trial, that set is said to be a sample space of that particular experiment. For example, when we flip a coin there are only two possibilities; either to get a head or a tail. If we represent head by ‘H’ and tail be ‘T’ then the sample space of the experiment of flipping a coin, say S = {H, T}.
To find the probability of any event associated with a random experiment we divide the number of outcomes of that event with the total number of possible outcomes of the whole experiment that is the sample space.

$P\ (event)$ = $\frac{number\ of\ outcomes\ of\ the\ event}{number\ of\ outcomes\ of\ the\ experiment}$

General Sample Space

In general sample space is denoted by three common symbols: U, S, $\Omega$.

Commonly we used, ‘U’ to represents universal set. An experiment can have more than one sample space depending on the condition been set.
For Example: if we see a deck of cards we have numerous possibilities of sample space for the event of choosing a card. A card can be drawn and classified on the basis of its suite then the sample space will be {diamond, club, heart, spade}, or it can also be classified on its number and then the sample space will be the ranks and cards from 2-9.

So we see that only on the basis of requirement in an experiment a sample space is set for that particular event of an experiment.

Sample Space Examples

Let us see some examples of finding the spaces for given experiments:
Example 1:

Let us suppose we are flipping three fair coins together. Find the sample space and favorable outcomes for at least two heads.

Solution:

Sample space, S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}

Favorable outcomes for at least two heads

= favorable outcomes for exactly two heads and favorable outcomes for more than two heads

= {HHH, HHT, HTH, THH}
Example 2:

Find the sample space for rolling two dices together and then find the probability of

i)   getting a total of 6

ii)   getting numbers in pairs

Solution:

Sample space, S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
N (S) = 36

Set of total of 6, A = {(1,5), (5,1), (2,4), (4,2), (3,3)}; N (A) = 5

P (total of 6) = $\frac{N (A)}{ N (S)}$ = $\frac{5}{36}$

Set of pairs, B = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}; N (B) = 6

P (getting numbers in pairs) = $\frac{N (B)}{N (S)}$ = $\frac{6}{36}$ = $\frac{1}{6}$
Example 3:

Find the probability of getting a club out of a deck of cards.

Solution: We have 52 cards in a deck, i.e. sample space will have 52 options so N (S) = 52.

Also, there are 13 cards of club in a deck so N (A) = 13 where A be the set of clubs.

So P (drawing a club from a deck) = $\frac{N (A)}{N(S)}$ = $\frac{13}{52}$ = $\frac{1}{4}$.