Sample Space

There are lots of events like when we toss a coin or toss a die, and we cannot predict the outcomes with certainty but we can always say all the possible outcomes. These events are what we call a random phenomenon or a random experiment. Probability theory is usually involved with such random phenomena or random experiments. But what is a sample space?  

A collection of a set of all such possible outcomes is known as a sample space of the experiment. In other words, we can say that in a random experiment, the collection or set of the possible outcomes is referred to as the sample space and is normally denoted by S. 

The sample space in case of a random experiment is represented within curly brackets “{ }.” The sample space may depend upon the number of outcomes in an experiment, and the subset of the possible outcomes is referred to as the event. If the number of outcomes is finite, then the sample space is known as discrete or finite sample space.

Now, we have two more questions. First, what is the probability? What are events? We got you! Here is the answer. 

Some Important Definitions

• Probability:

In mathematics, the probability is actually a branch that is concerned with numerical descriptions of how an event might occur or how it is that a proposition might be true. The probability of an event is basically a number between 0 and 1, where, on an estimate, 0 designates the impossibility of the event, and 1 designates certainty.

• Events:

Events are actually a subset of possible outcomes of an experiment.

Difference Between a Sample Space and an Event

Even though a sample space and an event are written within curly braces “ { } “, there is a difference between both of them. When we roll a die, we get sample space as  {1, 2, 3, 4, 5, 6 }, but an event will either represent a set of even numbers like  { 2, 4, 6 } or a set of odd numbers like  {1, 3, 5}.

Sample Space- Examples

1. Tossing a coin:

When we toss a coin, there can be only two outcomes i.e., either head or tail. So, the sample space will be, S = {H, T} where H is the head and T is the tail.

2. Tossing two coins together:

When we flip two coins together, we have a total of 4 outcomes. H1 and T1 can be represented as heads and tails of the first coin. H2 and T2 can be represented as heads and tails of the second coin. So, the sample space will be, 

S = {(H1, H2), (H1, T2), (T1, H2), (T1, T2)} 

With this, we know that if we have ‘n’ coins, the possible number of outcomes will be 2n.   

3. Rolling a dice:

On rolling a die, we can have 6 outcomes. So the sample space will be, S = {1, 2, 3, 4, 5, 6}.

4. Rolling Two Dice Together:

When we roll two dice together, we get double the outcomes than when we roll a single outcome. When we roll 2 dice together we get 36 outcomes (6 x 6 = 36).

So, the possible sample space will be, S = {(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). 

If we throw three dice together, we should have the possible outcomes of 216. Here n in the experiment will be taken as 3, so it becomes 63 = 216.

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In the same way, there are many events where we use the method of sample space such as while playing cards or even while deciding something. We can use it in many games too.

Sample Space- Solved Examples

1) What is the sample space for intervals [2,8]?

Solution 

Given Interval [2,8]

Since the given integers are in a closed interval, the value can be taken from 2 to 8.

Thus, the sample space for the given interval will be:

S={2,3,4,5,6,7,8}

2) What will be the probability of outcomes when we toss a dime?

Solution 2) The two outcomes of this experiment are heads and tails.

The probabilities will be: 

P(heads) = \[\frac{1}{2}\]

P(tails) = \[\frac{1}{2}\]

3) A spinner has 4 equal sectors that are colored yellow, blue, green, and red. What will be the probability of their landing on each color after we spin this spinner?

Solution 3) The sample space will be: 

S = {Red, Blue, Green, and Yellow}

So, the probability will be:

P (Red) = \[\frac{1}{4}\]

P (Blue) = \[\frac{1}{4}\]

P (Green) = \[\frac{1}{4}\]

P (Yellow) = \[\frac{1}{4}\]

4) What will be the sample space when two dies are thrown together?

Solution 4) There will be a total of 36 pairs of possible outcomes if two dies are thrown together since each of the faces of one die can fall with any of the six faces of the other die. The sample space can be given as 

S= { (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) }

Similarly, if the total number of dice thrown is 3, then the total number of possible outcomes would be 6 x 6 x 6 = 216.

5) If a coin is thrown thrice, what will be the sample space?

Solution 5) If a coin is tossed once, the number of possible outcomes are two which are head or tail thus the sample space is-

S = { H, T}

If a coin is tossed thrice, the number of possible outcome will be 2 x 2 x 2= 8,

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