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 sample space is a collection or a set of all such possible outcomes is known as a sample space of the experiment and is normally denoted by S. it may have a number of outcomes that usually depends on ob the experiment but if it has a finite set of numbers, then it is called a discrete or finite sample space.

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

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 & 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

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}

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.

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. And 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.

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}

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 1 dice, we get 6 outcomes and 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.

Question 1) What are the sample space for intervals [2, 8]

Solution 2) the sample space for the interval [2, 8] will be:

S = {2, 3, 4, 5, 6, 7, 8, 9}

Question 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) = 12

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

Question 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) = \[\frac{1}{4}\]

FAQ (Frequently Asked Questions)

1. What is a tree diagram?

When we attempt to decide a sample space, it has always proved helpful to draw a diagram that can illustrate how we can arrive at an answer.

The most basic example of one such diagram is a tree diagram. A tree diagram is a drawing that is drawn using "line segments" showing us all of the different possible "paths" for the outcomes.

Additionally, to assist in the determination of the number of outcomes in a sample space, the tree diagram can be useful to determine the probability of individual outcomes within the sample space.

In a sample space, the probability of any outcome is basically the product (multiplication) of all probabilities along a path that mostly represents the outcome on a tree diagram.

2. What is a random event?

A concept of an event is extremely significant in the Theory of Probabilities. Actually, it's one of the most fundamental concepts, just like a point in Geometry or equation in Algebra. First of all, we define a random experiment as any physical or mental act that has a certain number or set of outcomes. For example, we all count money in our wallet or maybe predict tomorrow's stock market index value. In both of these and many other cases, the random experiment results in certain outcomes (for example, counting the exact amount of money we have, the exact stock market index value, etc.)