Probability
Probability defines the likelihood of occurrence of an event. There are many reallife situations in which we may have to predict the outcome of an event. We may be sure or not sure of the results of an event. In such cases, we say that there is a probability of this event to occur or not occur. Probability generally has great applications in games, in business to make probabilitybased predictions, and also probability has an extensive application in this new area of artificial intelligence.
Here in this below content, we will discuss the basics of probability, terms used in probability, probability formula, conditional probability, the law of total probability, Bayes' theorem.
What is Probability?
Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having n number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.
Probability(Event) = Favourable Outcomes/Total Outcomes = x/n
Let us check a simple application of probability to understand it better If we have to check about the happening of rain or not. The answer to this question is either "Yes" or "No." There is a likelihood to rain or not rain. Here we can apply probability. Probability is used to predict the outcomes for the tossing of coins, rolling of dice, or drawing a card from a pack of playing cards.
Some important terms used in probability are as discussed below.
Terms in Probability
The following terms in probability help in a better understanding of the concepts of probability.
 Experiment: A trial or an operation conducted to produce an outcome is called an experiment.
 Sample Space: All the possible outcomes of an experiment together constitute a sample space. For example, the sample space of tossing a coin are head and tail.
 Favorable Outcome: An event that has produced the desired result or expected event is called a favorable outcome. For example, when we roll twodice, the possibile/favourable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).
 Trial: A trial denotes doing a random experiment.
 Random Experiment: An experiment that has a welldefined set of outcomes is called a random experiment. For example, when we toss a coin, we know that we would get ahead or tail, but we are not sure of which one will appear.
 Event: The total number of outcomes of a random experiment is called an event.
 Equally Likely Events: Events that have the same chances or probability of occurring are called equally likely events. The outcome of one event is independent of the other. For example, when we toss a coin, there are equal chances of getting a head or a tail.
 Exhaustive Events: When the set of all outcomes of an experiment is equal to the sample space, we call it an exhaustive event.
 Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either hot or cold. We cannot experience the same weather simultaneously.
Probability  Tossing a Coin
Let us now look into the probability of tossing a coin. Quite often in games like cricket, for making a decision as to who would bowl or bat first, we sometimes use the tossing of a coin and decide based on the outcome of the toss. Let us check as to how we can use the concept of probability in the tossing of a single coin. Further, we shall also look into the tossing of two and three coming respectively.
Tossing a Coin
A single coin on tossing has two outcomes, ahead, and a tail. The concept of probability which is the ratio of favorable outcomes to the total number of outcomes can be used to find the probability of getting the head and the probability of getting a tail.
Total number of possible outcomes = 2; Sample Space = {H, T}; H: Head, T: Tail
 P(H) = Number of heads/Total outcomes = 1/2
 P(T)= Number of Tails/ Total outcomes = 1/2
Tossing Two Coins
In the process of tossing two coins, we have a total of four outcomes. The probability formula can be used to find the probability of two heads one head, no head, and a similar probability can be calculated for the number of tails. The probability calculations for the two heads are as follows.
Total number of outcomes = 4; Sample Space = {(H, H), (H, T), (T, H), (T, T)}
 P(2H) = P(0 T) = Number of outcome with two heads/Total Outcomes = 1/4
 P(1H) = P(1T) = Number of outcomes with only one head/Total Outcomes = 2/4 = 1/2
 P(0H) = (2T) = Number of outcome with two heads/Total Outcomes = 1/4
Tossing Three Coins
The number of total outcomes on tossing three coins simultaneously is equal to 2^{3} = 8. For these outcomes, we can find the probability of getting one head, two heads, three heads, and no head. A similar probability can also be calculated for the number of tails.
Total number of outcomes = 2^{3} = 8 Sample Space = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}
 P(0H) = P(3T) = Number of outcomes with no heads/Total Outcomes = 1/8
 P(1H) = P(2T) = Number of Outcomes with one head/Total Outcomes = 3/8
 P(2H) = P(1T) = Number of outcomes with two heads /Total Outcomes = 3/8
 P(3H) = P(0T) = Number of outcomes with three heads/Total Outcomes = 1/8
Probability  Rolling a Dice
Many games use dice to decide the moves of players across the games. A dice has six possible outcomes and the outcomes of a dice is a game of chance and can be obtained by using the concepts of probability. Some games also use two dice, and there are numerous probabilities that can be calculated for outcomes using two dice. Let us now check the outcomes, their probabilities for one dice and two dice respectively.
Rolling One Dice
The total number of outcomes on rolling one dice is 6, and the sample space is {1, 2, 3, 4, 5, 6}. Here we shall compute the following few probabilities to help in better understanding the concept of probability on rolling one dice.
 P(Even Number) = Number of even number outcomes/Total Outcomes = 3/6 = 1/2
 P(Odd Number) = Number of odd number outcomes/Total Outcomes = 3/6 = 1/2
 P(Prime Number) = Number of prime number outcomes/Total Outcomes = 3/6 = 1/2
Rolling Two Dice
The total number of outcomes on rolling two dice is 6^{2} = 36. The following image shows the sample space of 36 outcomes on rolling two dice.
Let us check a few probabilities of the outcomes from two dice. The probabilities are as follows.
 Probability of getting a doublet(Same number) = 6/36 = 1/6
 Probability of getting a number 3 on at least one dice = 11/36
 Probability of getting a sum of 7 = 6/36 = 1/6
As we see, when we roll a single die, there are 6 possibilities. When we roll two dice, there are 36 possibilities. When we roll 3 dice we get 216 possibilities. So a general formula to represent the number of outcomes on rolling n dice is 6^{n}.
Probability of Drawing Cards
A deck containing 52 cards are grouped into four suits of clubs, diamonds, hearts, and spades. Each of the clubs, diamonds, hearts, and spades have 13 cards each, which sum up to 52. Now let us discuss the probability of drawing cards from a pack. The symbols on the cards are shown below. Spades and clubs are black cards. Hearts and diamonds are red cards.
The 13 cards in each suit are ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king. In these, jack, queen, king are called face cards. We can understand the probability of these playing cards from the following examples.
 Probability of drawing a black card is P(Black Card) = 26/52 = 1/2.
 Probability of drawing a hearts card is P(Hearts) = 13/52 = 1/4
 Probability of drawing a face card is P(Face card) = 12/52 = 3/13
 Probability of drawing a card numbered 4 is P(4) = 4/52 = 1/13
 Probability of drawing a red card numbered 4 is P(4 Red) = 2/52 = 1/26
Basic Theorems on Probability
The following theorems of probability are helpful to understand the applications of probability and also perform the numerous calculations involving probability.
Theorem 1
The sum of the probability of happening of an event and not happening of an event is equal to 1. \(P(A) + P(\bar A) = 1\)
Theorem 2
The probability of an impossible event or the probability of not happening of an event is always equal to 0. \(\begin{align}P(\phi) =0\end{align}\)
Theorem 3
The probability of a sure event is always equal to 1. P(A) = 1
Theorem 4
The probability of happening of any event always lies between 0 and 1. 0 < P(A) < 1
Theorem 5
If there are two events A and B, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event A or event B as follows.
\(\begin{align}P(A\cup B) = P(A) + P(B)  P(A\cap B)\end{align}\)
Also for two mutually exclusive event A and B we have P( A U B) = P(A) + P(B)
Bayes' Theorem on conditional probability
Bayes' theorem describes the probability of an event based on the condition of occurrence of other events. It is also called conditional probability. It helps in calculating the probability of happening of one event based on the condition of happening of another event.
For example, let us assume that there are three bags with each bag containing some blue, green, and yellow balls. What is the probability of picking a yellow ball from the third bag? Since there are also blue and green coloured balls, we can arrive at the probability based on these conditions also. Such a probability is called conditional probability.
The formula for Bayes' theorem is \(\begin{align}P(AB) = \dfrac{P(A) P(BA)} {P(B)}\end{align}\)
where \(\begin{align}P(AB) \end{align}\) denotes how often event A happens on a condition that B happens.
where \(\begin{align}P(BA) \end{align}\) denotes how often event B happens on a condition that A happens.
\(\begin{align}P(A) \end{align}\) the likelihood of occurrence of event A.
\(\begin{align}P(B) \end{align}\) the likelihood of occurrence of event B.
Law of Total Probability
If there are n number of events in an experiment, then the sum of the probabilities of those n events is always equal to 1.
\(P(A_1) + P(A_2) + P(A_3) + ....P(A_n) = 1\)
Related Topics
The following related topics help in better understanding of probability.
 Probability and Statistics
 Probability Rules
 Mutually Exclusive Events
 Independent Events
 Binomial Distribution
 Baye's Formula
 Poisson Distribution Formula
Important Notes
Let us check the below points, which help us summarize the key learnings for this topic of probability.
 Probability is a measure of how likely an event is to happen.
 Probability is represented as a fraction and always lies between 0 and 1
 An event can be defined as a subset of sample space.
 The outcome of throwing a coin is a head or a tail and the outcome of throwing dice is 1, 2, 3, 4, 5, or 6
 A random experiment cannot predict the exact outcomes but only some probable outcomes.
Solved Examples on Probability

Example 1: What is the probability of getting a sum of 10 when two dice are thrown?
Solution:
There are totally 36 possibilities when we throw two dice.
Desired outcome is 10. To get 10, we can have the following favorable outcomes.
\(\begin{align}(4,6),(6,4),(5,5)\}\end{align}\) . There are three favorable outcomes.
\(\begin{align} Probability\: of\: an\: event\:P(E)\:=\: \dfrac{Number\: of\: favorable\: outcomes}{Sample \: space}\end{align}\)
Probability of getting number \(\begin{align} 10\:=\: \dfrac{3}{36}\end{align}\)
which simplifies to
\(\begin{align} \dfrac{1}{12}\end{align}\)
Answer: Therefore the probability of getting a sum of 10 is 1/12.

Example 2: In a bag, there are 6 blue balls and 8 yellow balls. One ball is selected randomly from the bag. Find the probability of getting a blue ball.
Solution:
Let us assume the probability of drawing a blue ball to be P(B)
Number of favorable outcomes to get a blue ball = 6
Total number of balls in the bag = 14
P(B) = Number of favorable outcomes/Total number of outcomes = 6/14 = 3/7
Answer: Therefore the probability of drawing a blue ball is 3/7.
Frequently Asked Questions(FAQs) on Probability
What is Probability?
Probability is a branch of math which deals with finding out the likelihood of the occurrence of an event. Probability measures the chance of happening of an event and is equal to the number of favorable events divided by a total number of events. The value of probability ranges between 0 and 1.
How was Probability Discovered?
The use of the word probable started first in the seventeenth century when it was referred to actions or opinions which were held by sensible people. Further, the word probable in the legal content was referred to a proposition that had tangible proof. The field of permutations and combinations, statistical inference, cryptoanalysis, frequency analysis have altogether contributed to this current field of probability.
How to Determine Probability?
The probability can be determined by first knowing the sample space of outcomes of an experiment. A probability is generally calculated for an event (x) within the sample space. The probability of happening of an event is obtained by dividing the number of outcomes of an event by the total number of possible outcomes or sample space.
What Are the Three Types of Probability?
The three types of probabilities are theoretical probability, experimental probability, and axiomatic probability. The theoretical probability calculates the probability based on formulas and input values. The experimental probability gives a realistic value and is based on the experimental values for calculation. Quite often the theoretical and experimental probability differ in their results. And the axiomatic probability is based on the axioms which govern the concepts of probability.
What is Conditional Probability?
The conditional probability predicts the happening of one event based on the happening of another event. If there are two events A and B, conditional probability is a chance of occurrence of event B provided the event A has already occurred. The formula for the conditional probability of happening of event B, given that event A has happened is P(B/A) = P(A ∩ B)/P(A).
What is Experimental Probability?
The experimental probability is based on the results and the values obtained from the probability experiments. Experimental probability is defined as the ratio of the total number of times an event has occurred to the total number of trials conducted. The results of the experimental probability are based on reallife instances and may differ in values from theoretical probability.
What is a Probability Distribution?
The two important probability distributions are binomial distribution and poison distribution. The binomial distribution is defined for events with two probability outcomes and for events with a multiple number of times of such events. The poison distribution is based on the numerous probability outcomes in a limited space of time, distance, sample space. An example of the binomial distribution is the tossing of a coin with two outcomes, and for conducting such a tossing experiment with n number of coins. A poison distribution is for events such as antigen detection in a plasma sample, where the probabilities are numerous.
How are Probability and Statistics Related?
The probability calculates the happening of an experiment and it calculates the happening of a particular event with respect to the entire set of events. For simple events of a few numbers of events, it is easy to calculate the probability. But for calculating probabilities involving numerous events and to manage huge data relating to those events we need the help of statistics. Statistics helps in rightly analyzing
How Probability is Used in Real Life?
Probability has huge applications in games and analysis. Also in real life and industry areas where it is about prediction we make use of probability. The prediction of the price of a stock, or the performance of a team in cricket requires the use of probability concept. Further the new technology field of artificial intelligence is extensively based on probability.