what is probability Definition Formula Examples Tree Types Events

 What Is Probability?

The word probability means chance or possibility of an outcome. It explains the possibility of a particular event occurring. We often use sentences like - ‘It will probably rain today, ‘he will probably pass the test’, ‘there is very less probability of getting a storm tonight’, and ‘most probably the price of onion will go high again. In all these sentences, we replace words like chance, doubt, maybe, likely, etc., with the word probability.

Probability means that it is possible. It is a branch of statistics that deals with the occurrence of a random event. The number is expressed from zero to one. Our Probability is presented in Maths to predict how likely events will occur. 

Formula for Probability

 

P(E)=NumberoftrailsinwhichtheeventhappenedTotalnumberoftrails$$P(E)=\frac{Numberoftrailsinwhichtheeventhappened}{Totalnumberoftrails}$$

On the basis of the theoretical formula, we can say that the probability is:

P(E)=numberofoutcomesfavorabletoENumberofallpossibleoutcomesoftheexperiments$$P(E)=\frac{numberofoutcomesfavorabletoE}{Numberofallpossibleoutcomesoftheexperiments}$$

Example 1: What is the probability of getting a tail if a coin is tossed once?

Solution: The total number of possible outcomes is 2 that is Head and Tail.

Let the event of getting a tail be E.

The probability of getting a tail on tossing a coin is:  

P(E)=NumberofoutcomesfavorabletoENumberofallpossibleoutcomesoftheexperiments$$P(E)=\frac{NumberofoutcomesfavorabletoE}{Numberofallpossibleoutcomesoftheexperiments}$$

=

12$$\frac{1}{2}$$

Example 2: A bag contains a blue ball and a red ball and a yellow ball of the same size and weight. If Archana picks out a ball from the bag randomly, then what is the probability of getting an (i) blue ball (ii) yellow ball, and (iii) red ball. 

Solution:

The total number of balls inside the bag is 3 out of which one ball is red, one ball is blue, and yellow. If Archana takes out a ball from the bag randomly then 

(i) The probability of getting a blue ball = 1/3

(ii) The probability of getting a yellow ball = 1/3

(iii) The probability of getting a red ball = 1/3

Types of Probability

There are three main types of Probability:

1. Theoretical Probability

2. Probability to Test

3. Axiomatic Probability

Theoretical Probability

It is based on the probability that something will happen. Theoretical possibilities are primarily based on the concept of Probability. For example, if a coin is tossed, the chance of a head-turning theory will be ½.

Probability to Test

It is based on the basis of test recognition. Test scores can be calculated based on the number of possible results for the total number of tests. For example, if a coin is thrown 10 times and heads are recorded 6 times at a time, the probability of checking heads is 6/10 or 3/5.

Axiomatic Probability

With axiomatic possibilities, a set of rules or set axioms apply to all types. These axioms are set by Kolmogorov and are known as the three axioms of Kolmogorov. With the axiomatic approach to probability, the probability of occurrence or non-occurrence of events can be estimated. 

The axiomatic probability study incorporates this concept in detail with three Kolmogorov rules (axioms) and various examples.

Terms The conditions are the probability that an event or outcome will occur based on the occurrence of a previous event or outcome.

Event Probability

Assume that event E can occur in ways r without the sum of n possible or possible ways equally. Then the chances of an event or success being achieved are highlighted as;

P (E) = r / n

The chances of an event not occurring or being known as a failure are set out as follows:

P (E ’) = (n-r) / n = 1- (r / n)

E ’represents that the event will not take place.

So, now we can say;

P (E) + P (E ’) = 1

This means that the sum of all the possibilities for any random test or test is equal to 1.

Types of Event:

1. Complementary events

2. Independent events

3. Mutually exclusive events

Types of Probability

There are three major types of probabilities:

1. Theoretical Probability- Prediction about a particular event can be precisely made with the access of statistical data of an event. The definition of probability in statistics is based on the possibility of the occurrence of an outcome. Suppose if you are willing to find out the theoretical probability of getting a number '5' on rolling a die, then you should determine the number of possible outcomes. We are aware of the fact that a die has 6 numbers (i.e, 1,2,3,4,5,6), thus the number of possible outcomes is also six. So, the chance of getting 6 on rolling a die is one out of six; that is 1:6. Similarly, we know that the total number of possible outcomes on tossing a coin is 2 because you can either get your head or tail. Thus, the theoretical probability of getting head on tossing a coin is ½.

2. Experimental Probability- Experimental probability is the definition statistics of unlike theoretical probability definition includes the number of trials. Suppose a coin is tossed 30 times and out of those 30times, we got tails 12 times, then the experimental probability of getting ahead is 12:30. This calculation of probability is based on the prior carried out experiments. Experimental probability is equal to the number of all the possible outcomes of an event divided by the total number of trials. For example- a die rolled 50 times results in the appearance of 6 thrice. So the Experimental probability of getting six is 6/50. 

3. Axiomatic Probability- Axiomatic Probability is a theory of unifying probability where there is an application of a set of rules made by Kolmogorov. 

The three axioms are:

1. The probability of an event A is always greater or equal to zero but can never be less than zero.

2. If S is a sample space, then the probability of occurrence of sample space is always 1. That is, if the experiment is performed, then it is sure to get one of the sample spaces.

3. For mutually exclusive events, the probability of either of the events happening is the sum of the probability of both the events happening.

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