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## What is Probability?

Probability is a special branch of mathematics that deals with calculation of the likelihood of a provided occurrence of an event. The occurrence of an event is either represented as 0 or 1. Any event with probability 1 is a certainty. For example, the probability of a coin tossed can be either ‘head’ or ‘tail’ as there are no other options in this case, so the probability would be 1.

For the equal number of occurrences, .5 is considered as the probability. If the value is zero then this is the impossible situation in probability. Further, the probability is suitable for measurement of random events too.

## List of Basic Probability Formulas

Mathematically, Probability is defined as the number of occurrences for a targeted event plus the number of failure occurrences too. Based on the example given earlier, calculation of coin tossed is simpler because there are only two possible situations. But this is not true in every case where multiple datasets can be derived and they are impossible to solve without basic probability formulas.

0 ≤ P(A) ≤ 1

#### Rule of Complementary Events

P(AC) + P(A) = 1

P(A∪B) = P(A) + P(B) – P(A∩B)

#### Disjoint Events

Events A and B are disjoint iff

P(A∩B) = 0

#### Conditional Probability

P(A | B) = P(A∩B) / P(B)

#### Bayes Formula

P(A | B) = P(B | A) ⋅ P(A) / P(B)

#### Independent Events

Events A and B are independent iff

P(A∩B) = P(A) ⋅ P(B)

#### Cumulative Distribution Function

FX(x) = P(X ≤ x)

Commutative
$\ A\cup B = B\cup A$
$\ A\cap B = B\cap A$
Associative
$\ A\cup (B\cup C) = A\cup (B\cup C)$
$\ A\cap (B\cap C) = A\cap (B\cap C)$
Neutral element
$\ A\cup \theta = A$
$\ A\cap E = A$
Absorbing element
$\ A\cup E = E$
$\ A\cap \theta = \theta$
Distributive
$\ A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
$\ A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
De Morgan’s laws
$\ \bar(A\cap B) = \bar A \cup \bar B$
$\ \bar(A\cup B) = \bar A \cap \bar B$
Independent Events
$\ P(A | B)=P(A)$
$\ P(A\cap B)=P(A)×P(B)$
Conditional Probability
$\ P(A | B)=\frac{P(A\cap B)}{P(B)}$
Laplace laws
$\ P(A)=\frac{Number\;of\;ways\;it\;can\;happen}{Total\;Number\;of\;Outcomes}$
Complement of an Event
$\ P(\bar A)=1 – P(A)$
Union of Events
$\ P(A\cup B)=P(A)+P(B)−P(A\cap B)$
Permutation
$\ P_{n}= n! = n\times (n-1)\times . .\times 2\times 1$
Permutations without repetition
$\ ^nA_{p} = \frac{n!}{(n-p)!}$

### Probability Equation

The theory of probability was started during the 17th century by two French Mathematicians dealing with games of chances. The most common application of Probability is the game development of different categorize and especially the puzzle games. There is a list of formulas that can be placed in equations to solve any expression. There are special computer programs to deal with complex probability equations.

### Probability Problem with Solution

In simple words, Probability is the term used to define the randomness. If there is some problem related to randomness or to find the maximum number of occurrences then probability works best in this situation. With the same objective, it was added to students’ curriculum so that they can understand the random behavior of objects and find the solution of difficult probability problems.

Steps to find the probability:

Step 1: List the outcomes of the experiment.

Step 2: Count the number of possible outcomes of the experiment.

Step 3: Count the number of favorable outcomes.

Step 4: Use the probability formula.

#### Question 1: Calculate the probability of getting an odd number if a die is rolled?

Solution:

Sample space(S) if a die is rolled = {1, 2, 3, 4, 5, 6}

Let “E” be the event of getting an odd number, E = {1, 3, 5}

So, the Probability of getting an odd number P(E) = Number of outcomes favourable / Total number of outcomes = n(E) / n(S) = 3 / 6 = 1 / 2

#### Question 2: If two coins are tossed, then calculate the probability of getting two tails?

Solution:

Sample space(S), when two coins are tossed = {(H, H), (H, T), (T, H), (T, T) } = 4

Let “E” is the event of getting an odd number, E = {(T, T)} = 1

So, the Probability of getting an odd number P(E) = Number of outcomes favourable / Total number of outcomes = n(E ) / n(S) = 1 / 4

### Why do we need Probability Formula for Students?

A deep knowledge of probability is necessary for understanding measurements, numbers etc. that are further necessary to pursue science or engineering studies. So, you need to honest yourself and judge deeply where you stand in probability and improve your skills accordingly. You just don’t need to study Probability but master the skills to become a pro.

When one is interested in knowing the thoughts of a growing child then this is necessary to make him logically stronger and tempt with all the skills in advance. It includes the teaching of experimental probability too that makes him a skilled professional in later life. Additionally, Probability allows students making sense of experiences that involves chances.

Further, understanding of probability is necessary to work on things like weather reports, sports, genetics, or insurance policies etc. There is a special type of experimental probability that represents outcomes of a trial or experiment. So, now you understand why probability formulas are needed for students and how they can help them in progressing there career tremendously. Mastering probability skills are not always easy. So, you should join some online course to get the best results.