Probability in Statistics: Formulas, Theory & Examples

Probability in Statistics expresses the chance of an incident occurring. The value of probability ranges from zero to one. There are several circumstances in which we would predict the outcome of an event in real life. We may be certain or uncertain about the outcome of an event. In such instances, we believe that there is a probability that this event will happen.

Probability is the branch of mathematics, which refers to the occurrence of a random experiment. Probability refers to possibility.  In this article, we will provide all the detailed information about probability, its formula, types, solved examples, etc. Scroll down to continue reading!

What is Probability?

Probability is the branch of mathematics, which discusses the occurrence of a random experiment. Probability means possibility.  Since many events cannot be predicted with absolute certainty, probability helps to predict the likelihood of an event to occur. The measurement of the possibility of an event is called probability.

Probability Formula

The probability formula is the ratio of the number of favourable events to the total number of events in an experiment.

Numerically, the probability value always lies between \(0\) and \(1\).

Probability is expressed in decimal, percentage, or a fraction and it cannot be a negative value.

Types of Probability

There are mainly \(3\) types of probabilities:

1. Theoretical Probability
2. Experimental Probability
3. Axiomatic Probability

1. Theoretical Probability

Theoretical probability is based on the possible chances of something happening. It is based on what is expected to happen in an experiment without conducting it. It is the ratio of the number of favourable outcomes to the total number of outcomes.

2. Experimental Probability

Experimental probability is a probability that is determined based on a series of experiments. Therefore, it is based on the data which is obtained after an experiment is carried out. It is the ratio of the number of times an event occurs to the total number of experiments that are conducted.

3. Axiomatic Probability

In axiomatic probability, a set of rules or axioms are set, which applies to all types. In this probability, the chances of occurrence and non-occurrence of the events can be quantified. It is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

Terms Related to Probability

There are so many words associated with the definition and formula of probability. Some of the terms listed below:

1. Experiment: An activity whose outcomes are not known is an experiment. Every experiment has few favourable outcomes and few unfavourable outcomes.
2. Random Experiment: A random experiment is an experiment for which the set of possible outcomes is known.
3. Trial: The various attempts in the process of an experiment are called trials.
4. Event: A trial with a well-defined outcome is an event.
5. Outcome: It is the result of a trial or an experiment.
6. Sample Space: It is the set of outcomes of all the trials in an experiment.
7. Sample Point: It is one of the possible results of a trial.

What are Events in Probability?

In probability, an event is an outcome or defined collection of outcomes of a random experiment. Since the collection of all possible outcomes to a random experiment is called the sample space, another definition of an event is that it is any subset of a sample space. The likelihood of occurrence of an event is called probability.

There are many events associated with the sample space. Some of the important events are listed below:

1. Equally likely Events

When the events have the same probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring.

The following are some examples of likely outcomes:

1. Getting \({\rm{3}}\) and \({\rm{5}}\) on throwing die
2. Getting an even number and an odd number on a die.

2. Complementary Events

Complementary events occur when there are just two events, and one event is exactly opposite to the other event. Hence, \(A \cup \bar A = {\rm{set}}\,{\rm{of}}\,{\rm{sample}}\,{\rm{space}}\)

For an event with probability \(P(A)\), its complement is \(P(\bar A)\) such that
\(P(\bar A) + P(A) = 1\)

Example: In an examination, the event of success and the event of failure are complementary events.
\(P{\rm{(Success)}} + P{\rm{(Failure)}} = 1\)

3. Impossible Events

The event that cannot happen is called an impossible event. An event that is not a part of the experiment or which does not belong to the sample space of the outcomes of the experiment can be referred to as an impossible event. The probability of an impossible event is zero \({\rm{(0)}}\)

Example: Probability of getting number \(8\) on throwing a single dice.

4. Mutually Exclusive Events

Two events such that the happening of one event prevents the happening of another event are referred to as mutually exclusive events. In other words, two events are said to be mutually exclusive events if they cannot occur at the same time.

Example: Tossing a coin can result in either heads or tails, but both cannot be seen at the same time.

What are Experiments in Probability?

Below we have listed the events that are called experiments in Probability:

1. Tossing a Coin

Probability of getting head \( = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{heads }}}}{{{\rm{ Total}}\,{\rm{Outcomes }}}} = \frac{1}{2}\)

Probability of getting tail \( = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{tails}}}}{{{\rm{ Total}}\,{\rm{Outcomes }}}} = \frac{1}{2}\)

2. Throwing a Dice

There are total \(6\) outcomes while throwing dice. They are \(1,\,2,\,3,\,4,\,5,\,6.\) The probability of getting each number while throwing dice is equally likely outcomes as their probability equals to \(\frac{1}{6}\).

\({\rm{Probability}}\,{\rm{of}}\,{\rm{getting}}(1,2,3,4,5,6) = \frac{1}{6}\)

3. Drawing Cards

A deck containing \(52\) cards, that are grouped into four suits of clubs, diamonds, hearts and spades. 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, while hearts and diamonds are red cards.

In each suite, there is an ace, king, queen, jack \(10,\,9,\,8,\,7,\,6,\,5,\,4,\,3,\,2.\) We can apply the same formula of probability to find the probability of drawing a single card or two or more cards.

What is Probability Theory?

Probability theory is the branch of mathematics that deals with the possibility of the happening of events. Although there are many distinct probability interpretations, probability theory interprets the concept precisely by expressing it through a set of axioms or hypotheses.

These hypotheses help form the probability in terms of a possibility space, which allows a measure to hold values between \(0\) and \(1\). This is also known as the probability measure, to a set of possible outcomes of the sample space.

All Probability Formulas

Let \(A\) and \(B\) are two events. The probability formulas are listed below:

What is Probability Density Function?

The probability density function (PDF) is the probability function that is represented for the density of a continuous random variable lying between a certain range of values. The probability density function explains the normal distribution and how mean and deviation exist.

The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables whose distribution is not known.

Solved Examples – Probability

Q.1. Calculate the probability of getting an even number if a dice is rolled.
Ans: Sample space \(\left( S \right) = \left\{ {1,\,2,\,3,\,4,\,5,\,6} \right\}\)
Total number of outcomes \(n\left( S \right) = 6\)|

Let \(E\) and be the event of getting an odd number. \(E = \{ 2,4,6\} \) and \(n \in  = 3\)
So, the Probability of getting an odd number is:
\(P(E) = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{out}}\,{\rm{comes}}\,{\rm{favorable }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{out}}\,{\rm{comes }}}}\)
\( \Rightarrow P(E) = \frac{{n(E)}}{{n(S)}} = \frac{3}{6} = \frac{1}{2}\)

Q.2. Keerthi is picking a vowel from the set of English alphabets. What is the probability of taking the vowel from the alphabets?
Ans: Total number of English alphabets are \(26\).

Set of vowels are \(a,\,e,\,i,\,o,\,u\).
The total number of vowels are \(5\).
Probability of getting a vowel \(P(E) = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes }}}} = \frac{5}{{26}}\)

Q.3. Asit takes two coins and flips them both at once. What is the probability of getting heads on both coins?

Ans:

Sample space on flipping two coins \( = \{ (H,H),(H,T),(T,H),(T,T)\} \)
The total number of outcomes is \(4\).
The favourable outcomes of getting heads on both the coins is \(1\)
Probability of getting two heads \( = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{outcomes}}\,{\rm{witht}}\,{\rm{woheads }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes }}}}\)
\( \Rightarrow P(E) = \frac{1}{4}\)

Q.4. If \(P(E) = 0.05\) what is the probability of ‘not \(E\)’ ?
Ans: Given \(P(E) = 0.05\),
We know that \(P(\bar E) + P(E) = 1\)
Probability of not \(E,P(\bar E) = 1 – P(E)\)
\( \Rightarrow P(\bar E) = 1 – 0.05 = 0.95\)

Q.5.A bag contains \(3\) red balls and \(5\) black balls. Avinash is selecting a ball randomly from the bag. What is the probability that a ball drawn is red?
Ans: Given, the number of red balls in a bag \(3\).
The number of black balls in a bag \(5\).
The total number of balls in the bag \(3\, + \,5\, = \,8\).
Probability of getting red balls: \( = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{red}}\,{\rm{balls }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of balls }}}}\)
\( \Rightarrow P(E) = \frac{3}{8}\)

Q.6. One card is drawn at random from the well-shuffled deck of \(52\) cards. Find the probability of getting (a) A king of red colour (b) A faced card
Ans: Total number of cards in a deck are \(52\).

(a) Number of red colour kings are \(2\).
Probability of getting red colour king is given by:
\(P(E) = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{red}}\,{\rm{colour}}\,{\rm{kings }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{cards }}}} = \frac{2}{{52}} = \frac{1}{{26}}\)
(b) Number of faced cards in the deck are \(12\).
Probability of getting faced cards is given by:
\(P(E) = \frac{{{\rm{ Number}}\,{\rm{of}}\,{\rm{faced}}\,{\rm{cards }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{cards }}}} = \frac{{12}}{{52}} = \frac{3}{{13}}\)

Summary

Probability can also be defined as a possibility. Probability is the branch of mathematics that deals with the occurrence of a random event. The probability formula is the ratio of the number of favourable events to the total number of events in an experiment. The probability can be classified into 3 types, namely, Theoretical probability, Experimental probability, and Axiomatic probability.

In this article, we have explored the topic of probability, by knowing the definition of probability, the important terms in probability, by solving the examples and the interactive problems. There are many real-life situations in which we may have to predict the outcomes 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.

FAQs on Probability

Below here we have provided some frequently asked questions related to Probability:

Q.1. How do you calculate probability?
Ans: Probability can be calculated by using the formula, which is the ratio of the number of favourable events to the total number of events in an experiment.
\({\rm{Probability(Event) = }}\frac{{{\rm{ Favourable}}\,{\rm{outcomes }}}}{{{\rm{ Total}}\,{\rm{number}}\,{\rm{of}}\,{\rm{outcomes }}}}\)

Q.2. What are the 3 types of probability?
Ans: There are mainly 3 types of probabilities:
1. Theoretical Probability
2. Experimental Probability
3. Axiomatic Probability

Q.3. What is probability? Explain with an example.
Ans: Probability defines the likelihood of occurrence of an event. The measurement of the possibility of an event is called Probability. It is the ratio of the number of favourable outcomes to the total number of outcomes. For example:

1. Getting \(3\) and \(5\) on throwing a die
2. Getting an even number and an odd number on a die

Q.4. What are the 5 rules of probability?
Ans: The five rules of probability are:

1. The probability value always lies between \(0\) and \(1\). \(0 \leqslant P(A) \leqslant 1\)
2. The sum of the probabilities of all possible outcomes of an experiment is equal to
   \(P\left( {{A_1}} \right) + P\left( {{A_2}} \right) + P\left( {{A_3}} \right) \ldots ..P\left( {{A_n}} \right) = 1\).
3. The sum of probability of an event and its complementary event is equal to \(1\). \(P(A) + P\left({{A^ – }} \right) = 1\).
4. The sum of the probabilities of two events \(A\) and \(B\) is \(P(A \cup B) = P(A) + P(B) – P(A \cap B)\).
5. The product of the probability of independent two events \(A\) and \(B\) is \(P(A) \cdot P(B) = P(A \cap B)\)

Q.5. What is the probability of a sure event?
Ans: The probability of a sure event is one.
\(P(E) = 1\)

Q.6. Can a probability be negative?
Ans: The probability value of the event can not be negative. It is a positive value between \(0\) and \(1\).

Q.7. What is the probability of an impossible event?
Ans: The probability of an impossible event is \(0\).