Probability Maths Articles

  Probability is an expectation or assumption about occurring of event. It tells about the measure whether an event will occur or not. Suppose, a coin is tossed then the assumption or expectation of the occurrence of head or tail is termed as probability and the probability of the occurrence of head and tail will be equal that is two.

Bayes theorem can be defined as a theorem which has different interpretations. It concerns two events and relates inverse representations of probabilities concerning these events. Suppose, the occurrence of probability of a and the probability of b are given and the occurrence of b when a has already occurred is also given, now we need to determine the probability of a when b has already occurred is represented by P(a/b). Then here the bayes theorem is used and it will be determined as P(a/b) will be equals to P(b/a) multiplied with the probability of a that is represented as P(a) and divided with the probability of b that is represented as P(b). Thus the whole formula can be represented as P(a/b) = [P(b/a) P(a)]/ P(b). If in any problem, more than two variables are given then it will also be used in the bayes theorem.

In general terms, P(a/b) can be equated as P(a ᴧ b) / P(b) where P(b) must not be equals to zero in order to avoid the infinite occurrence of error. Similarly, P(b/a) will be equals to P(a ᴧ b) / P(a) if P(a) is not equals to zero. Bayes rule for three or more occurrence of events can be given as P(a| b ᴧ c) will be equals to probability of the intersection of all three events that is represented as P(a ᴧ b ᴧ c) divided by P(b ᴧ c). It can also be further equated as [P(c|a ᴧ b) P(a ᴧ b)] divided by P(c|b) P(b). For understanding the topic make a bar graph.

In the previous post we have discussed about How to Find Variance and In today's session we are going to discuss about bay s theorem. Hello friends, in this blog we will discuss about bay s theorem. In mathematics, bay’s theorem can have two distinct interpretations. It is a basic concept of Bayesian statistics and have different properties in the field of science and engineering. It is first discovered in the year 1701 – 1761 by the great scientist Thomas Bayes. The formula used for finding the probability that named was given after the 18 th – century by the great scientist British mathematician Thomas bayes. This theorem shows the way to revise existing terms or theories that was got new or additional evidence. If we take the case of finance then this theorem is used to see the risk of lending money to potential borrowers.

Now we will see the formula used in Bay’s theorem is given as:

U (S / T) = U (S ∩ T) / U (T) = U (S) * U (T | S) / U (T)

It is also said to be bayes rule.

Now we will see the the base theorem: Application used in bayes theorem are spread all over in mathematics, and are not limited to the financial realm. Basically the bayes theorem is used find the accuracy of medical test as a result by taking into accumulation that how likely any person or thing is to have a disease and the general accuracy of the test.

Let’s see the extended form of bayes theorem. It can be written as:

U (T) = ∑j U (T | Sj) U (Sj) it can also be written as:

U ( Sj | T)= U (T | Sj) U (Sj) / ∑ j U (T | Sj) U (Sj)

In the special cases it can be written as:

U (S | T) = U (T | S) U (S) / U (T | S) U (S) + U (T | ⌐ S) U (⌐ S). This is all about bayes theorem.

In the previous post we have discussed about basic operations on sets and In today's session we are going to discuss about How to Find Variance. The variance in the mathematics may be defined as the average of all the squared differences of the numbers given in the data, calculated from the mean. So here we will discuss about how to find variance.

The calculation of the variance just includes some very simple steps which are given as follows. First of all we have to calculate the arithmetic mean of all the numbers given in the data which is nothing but the average of all those numbers as calculated generally. Then the second step is to calculate the difference of every number from the calculated mean that is number minus the mean. Then we have to square all those differences calculated in the above step. The next and the final step is to calculate the average of all those squared differences.

Now let us take a very simple example to understand very precisely about how to find variance of any given data.

Suppose the numbers given in the data are 7, 3, 12, 5 and 8. Then first of all we have to calculate the arithmetic mean of all these numbers that is their average. (know more about How to Find Variance, here)

Thus, mean = ( 7 + 3 + 12 + 5 + 8 ) / 5 = 35 / 5 = 7.

Now we have to calculate the difference of each number from this mean which comes out as 0, -4, 5, -2 and 1. Now we have to square all these differences which will give 0, 16, 25, 4 and 1. The final step is to calculate the average of all these squared differences to get the variance.

Thus, variance = ( 0 + 16 + 25 + 4 + 1 ) / 5 = 46 / 5 = 9.2

Set can be defined as a gathering of well defined and dissimilar objects. It is one of the most useful concept or deal with the mathematics. Now we will talk about the basic operations on sets. There are different types of set are defined which is shown below.

1.      Union of set

2.      Intersection of sets

3.      Complement of sets

4.      Cartesian product of sets

Let’s have small introduction about all the sets. (know more about Set , here)

Union of sets: - If we have two sets (P and Q) and the addition of both the sets with together is said to be union of sets. The union of two set P and Q are represented by the P ∪Q. It means the set of all objects that are the member of either the set P or Q.

Intersection of sets: - Let we have two sets P and Q then the intersection of two set can be represented by P ∩ Q, it means the set of all objects or things that are the member of both the objects. If P ∩ Q = ⊘, then the set P and Q are said to be disjoint set.

Compliment of sets: - If we have two sets (P and Q) and the set of all the elements that are the member of set P but are not the member of set Q. Compliment of two sets can be represented by P Q or it is also represented by (P – Q).

In the previous post we have discussed about converge in probability and In today's session we are going to discuss about Permutation, It is stated as an arrangement of the objects. When we talk about the arrangement of things it will describe as an ordering in which things are organize according to a plan.

Permutation is a part of statistics that depict the set up of things put into the array. We can explain it as if there is a set S of elements as a , b , c …. then according to the permutation , elements of set S express as one to one correspondence. That means values are arranged from set S to set S , we can formally say it , permutation define the one to one correspondence of set to itself as if there is a set S then according to the permutation function f : S → S.

For describe permutation we can take an example as in the given array of elements there are two numbers they can be arranged as first put the first number and then second and in next arrangement put second value first and then first value. So permutation is only provide the way of arranging values.

We can find the number of subset that will created by the permutation using the formula as:

v p k = v ! / ( v – k) ! .

In the above formula v gives the number of element in the set and k give number of elements in the ordered subset.

As in the given example if there is 4 elements as (a ,b ,c ,d) then find the number of subset having two elements in each subset then put the values in the above given formula 4 ! / (4 – 2) ! that is 4 ! / 2 ! = 12 that means there is total 12 subset are created having two elements.

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In the previous post we have discussed about Solve Permutation Combination Problems and In today's session we are going to discuss about converge in probability, There are many terms under the theory of the probability in the subject of the math which we generally study. Some of the terms are very common which most of us would have come across for sure but there are also some terms which are not very common like the term convergence. So in this article we will try to go through some of the important facts about the convergence in probability.

In the theory of the probability we have many different types of the notions of the convergence of the random variables. The convergence of the various sequences of the random type of the variables to a limit the random variable can be considered as a very important and a useful concept in the theory of the probability and it also has many applications in the theory of the statistics.

The most basic fact behind the convergence in probability can be given as follows that is the probability of any outcome which is unusual tends to become lesser and lesser when the sequence is progressed.

This theory of the convergence in the probability is also utilized very commonly in the statistics. Let us take an example to understand this thing more precisely. We can take an example that an estimator is considered to be consistent when it converges in the probability to a quantity which is just estimated. The convergence in probability can also be considered as the kind of the convergence which is established with the help of the weak law of the large numbers. An interesting property of the convergence in the probability that is it implies the convergence in the distribution.

Probability is a concept which is used for solving problems that are related to mathematics. Probability is a way to calculate all possible occurring events a/ b when an event has ‘a’ ways to perform and result are obtained in ‘b’ ways. The probability of an event can generally be expressed in the form of range from 0 to 1.

Here we are going to discussing about permutation combination that basically deals with arranging the data values in either ordered form or non order form. Suppose we have different items like d, then the no. of ways to arranging then in a row can be calculated by the factorial d!. In the simple mean we can say that permutation can be consider as a arranging a thing where the ordering of things is considered. In the same aspect combination can be referred as selection of things in which ordering of things doesn’t matter.

In mathematical definition we can say that a permutation is a term which performs the sequencing of numbers by following the finite set order without using any repeated elements. Through this definition we can say that the concept of permutation is similar to the concept of factorial of any number. On the other side the concept combination can be describe as a way of selecting different things without following any specific order. When we simply want to select the total number of possible combination then it can be performed by using multiplication.  

In the previous post we have discussed about Sample Space and In today's session we are going to discuss about How to tackle Conditional Probability. In this blog we are going to discuss about the conditional probability that is define as the probability of the occurrence of event q that tell about the event p that is already occurred. This probability is expressed as P ( q / p) where p and q are defining the events and P define the probability that is known as the conditional probability. These events are independent to each other and events have no effect on each other. In the conditional probability the event occurrence of q gives the probability of occurrence of event p that is expressed as p (q).I this case when the event p and q are not independent then the probability of events p and q are define as the intersection of the p and q as

(p ans q) = P (p) P ( q/ p)

and by using the above define definition we can easily obtain the conditional probability of the events p and q as P (q / p) = P ( p and q) / P (p).  (know more about Conditional probability, here)

We can define it by a simple example as in a game of cards if a player wants to draw the cards of two spades in total of 52 cards then find the probability of drawing the cards?

Solution: As we know that in the stack of cards there are total 52 cards and there are total 13 cards of spade in the stack so when player first time draw the cards having the 13 cards in second chance he have only 12 cards in the total of 51 cards so there is conditional probability is define as the

p ( draw the second heart card / total remaining cards) = 12 / 15 .

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In the previous post we have discussed about How to Find Means and In today's session we are going to discuss about Sample Space In mathematics, the study of probability can be considered as a branch of mathematics that deals with the calculation of events which are occurred during an experiment. In mathematics, the probability of any particular event can be expressed in the form of number from zero to one. Any event which has the probability equals to one then that event consider as certain event. On the other hand probability equals to zero for an event can be consider as impossible to occur. The probability for an event depends on the sample data that are obtained from several experiments. According to the standard definition of probability we can say that sample space is the set of all possible outcomes. It means to say that it contains all possible outcomes which may be occurred during performing an experiment or an event. (know more about Sample space , here)

We can understand the importance of probability by performing an experiment on tossing a coin. Suppose we toss a coin which has two faces that is head and tail. When we toss two coins simultaneously two times then result would be two tails or zero tail or may be one tail. Then by using the concept of sample space we can simply perform the probability of an event very easily. In case of tossing a coin we can make a sample space as [TT, TH, HH, HT ]. The generated sample space contains all the possible outcomes that may be occurred during a tossing two coins. The sample space is a better concept in comparison of other concept of probability because each event is equally likely to occur as any other.

In the previous post we have discussed about Explain permutation and combination and In today's session we are going to discuss about How to Find Means. In this blog we are going to learn about the mean. In statistics mean is define the average value for the given set of numbers .These numbers are in arbitrary form means these numbers are not in the specific order. The mean value for the given set of numbers is defined as average value in general. Usually we calculate the arithmetic mean of the numbers that are given in form of average value of these numbers. (know more about Means, here)

There is also one other type of mean that is known s the populated mean that define the expected value for the defined random variable.

When we talk about the arithmetic mean it is define the average as follows:

If there are given a data set as p1, p2, p3, p4 and p5 then the arithmetic mean is calculated by using the formula as

Average (arithmetic mean) = sum of all the numbers in the data set / number of values

So according to the formula mean = p1 + p2 + p3 + p4 + p5 / 5

In statistics there are mainly three types of mean values are defined that are as follows:

(1): Arithmetic mean

(2): Geometric mean

(3): Harmonic mean

In geometric mean we can calculate the mean value for the given data set by multiplying the number rather than adding these numbers .there are basically define the values that are define in terms of rates as birth rate or mortal rate or growth rate etc.

Geometric mean = (p1 * p2 * p3 * p4 * p5) > 1/5

Harmonic mean is used to define the average value in terms of determining the relation with respect to some unit as acceleration that is defined as speed in per unit time.

In the previous post we have discussed about Free Online Tutor Help in solving Math Problems and In today's session we are going to discuss about permutation and combination, It is slightly different meaning from its original meaning. Permutation and combination has separated meaning and we can easily distinguish them on the orders by the arrangement of the objects. Combinations are not described by any specific order because in combination there exist no specific order while in permutation order is required.

Permutation can be defined as collection of objects by arranging them in the proper order.  We can also prove this with the help of an illustration like ST≠TS, we need one more example to understand this, consider any three names of your friends I have consider three names of my friend Maria (m), George (G), Symonds(S). Now the problem arise is that in how many possible ways these names can be arranged. I have supposed their names as MGS or you can even consider it in the other abbreviated form.  Now we will arrange them in the possible number of orders, so our concluded outcome will be MGS, GMS, GSM, SGM, SMG, MSG

So here the possible number of count of their names is 6. There is also the other way to find the possible number of counts this is with the help of the formula:

n! = (n) (n-1) (n-2). . . . Here the symbol ‘!’ is of factorial.

Now we will insert our values in the formula of whose result we want, so we need the result of three persons therefore our solution will be

3! = (3) (2) (1) = 6.

Now coming to the combination we already mentioned that in combination we are not required the arrangement in a particular order. Therefore here the example ST=TS , and even if we take the example of more numbers it will not hamper the arrangement .here if we want to calculate more numbers suppose we want to arrange the 6 persons in the group of three then we can easily do it with the help of combination formula which is given below

C (n, r) = n! / r!(n - r)!,

Although the order does not matter so we can find this easily C (6, 3) = 6! / 3! (6 - 3)! = 720 / 36 = 20.

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When we talk about the probability worksheets grade 3, we say that the probability is the concept dealing with the possibility of any event to occur. Probability was initially started in old times for playing the games related to the possibility of any event to happen. It included all the games played with dice, and gambling. But now the scope of probability has extended in our day to day life. In modern times we do some or the other predictions of any event in the real life situation and then we come to certain conclusions based on those predictions and analysis. Thus we say that the game of probability has started playing a major role in our day to day life. Thus we can see the study of probability has become important and extended to a wider extend. Now if we look at the words may occur, possibility or uncertain, these words indicate that if any of the experiment is performed, then the possibility of the particular result may be or may not be. In such case the chances of certainty reduces and the role of probability increases.  Suppose early morning a farmer predicts that there is the possibility of the rainfall to occur, then we say that the rainfall may occur or may not occur. Thus we say that the probability   is used now days in different fields of economics, business, and sciences. The study of probability is separately studied in details in the statistical study of any data. The games of coins, cards and dice include the study of probability of different combinations of the results, which comes in the study of statistics mathematics.

Previously we have discussed about differential equations formulas and In today's session we are going to discuss about Empirical Probability Formula and its a part of andhra pradesh secondary board,  By empirical probability, we mean that probability of an event represents the proportion of times and that too under the identical circumstances; the outcome can be expected to occur. The result of empirical probability refers to the long run frequency of occurrence. But before we study empirical probability definition, we take the following assumptions: a) The observations collected are random

b) We have large number of observations.

 Now we define empirical probability as follows “If the experiment is repeated for a large number of times under essentially identical conditions, then the limiting value of the ratio of the number of times of event ‘A’ happens to be the total number of trials of the experiments as the number of trials increases uncountable number of times, is called the probability of the occurrence of ‘A’ event “

This study of probability is also called the statistical probability.

 Let us assume that the event A takes place m number of times in ‘N’ repetition of the random experiment. Then we can find the relative frequency of the event ‘A’ by the formula m/ N and this will not change from one trial to another. When we observe that the value of ‘N’ increases infinitely, then it will more or less settle to the number which we call as the probability of ‘A’ event.

 We call this probability as the statistical probability since it is the relative frequency approach; the probability is obtained objectively by repetitive empirical observations of the statistical data. We must remember that the empirical probability P (A) can never be practiced because the relative frequency m/N may not attain a unique value, no matter how ever large ‘N’ may be. Also experimental conditions may not remain same and identical in all the situations if we have large number of experiments. 

Previously we have discussed about subtracting integers worksheet and In today's session we are going to discuss about Complimentary Events which comes under state secondary board andhra pradesh, In the simple definition we can define the probability as something will happen. It means to say that before performing any task we predict that what will happen. This process is known as probability. In a more standard definition we can say that probability of an event is the process of finding the number of favorable and unfavorable cases in the event. Suppose we toss a coin at that time we can say that the output either is a head or tail but not both. If we toss a coin hundred then before tossing a coin we can easily perform the probability for output.

 Here we are going to discussing about the complementary events in probability theory. In probability theory complimentary events of an event x can be define as inverse of not x. it means to say that complimentary of an event x contains all the events that are not exists in probability of event x. according to the probability theory, compliment of event x and event x are exhaustive in nature. In the standard definition we can define the complimentary events as “The complement of an event p contains all the set of outcomes in the sample space that are not included in the outcomes of event p.

The complement of event p in probability theory can be represented by given format p’.

In the below we show the example that helps you to understand the concept of complementary events in probability.

Example: what is the probability of selecting a card that is not a king? There is standard deck of card available that contains the 52 cards into it.

Solution: as we know that there are 4 king exists into the deck of cards then probability can easily be calculated by finding the complement of the pack of card. Let’s show you below:

P’ (king’) = 1 – P (king)

                 = 1 – 4 / 52

                 = 48 / 52

                = 12 / 13

So, we can see that by using complement of an event the probability for selecting a card that should not be a king is 12 / 13.

Previously we have discussed about first order differential equation and In today's session we are going to discuss about Addition Theorem in probability which is a part of cbse books for class 11, In mathematics, probability can be considering as a branch of mathematics which is popularly used for estimating the likelihood occurrence any particular event. Basically the concept of probability is used for predict the output of any particular event which are performed again and again. The probability is a mathematical term which can be express between the ranges from 0 to 1. if in case the probability of an event contains the value o then we can say that the possibility of that event is impossible to done. In the same aspect if probability of an event is 1 then we can say that the event is in a certain status. An event with a probability of .5 can be considered to have equal odds of occurring or not occurring.

Here we are going to discussing about the addition theorem in probability. According to the Addition theorem of probability, the probability of an event x and y can be determine by the probability that event A or event B occurs or both occur. In the simple mean we Define Addition Theorem as for any two events like

 x and y the probability of x union y is equals to the probability of x added to probability of y then minus the probability of x intersection y from them.

P (x ∪y) = P( x ) + P( y ) - P(x ∩ y )

The above notation Define Addition Theorem in the form of mathematical notation: P(x ) = It describe the probability of an event x occurs.

P(y) = It describe the probability of an event y occurs.

P(x U y ) =  it describe the probability that event x or event y occurs. P(x ∩ y )= it describe the probability that event x or event y occurs. The addition theorem probability can be proved by the following process, which are given as: For mutual exclusive events, the events which cannot occur together: P(x ∩ y )= 0  The addition rule therefore reduces to P(x U y ) = P( x ) + P( y )

On the other side for any independent events, which have no influence on each other 

P(x ∩ y )= P( x ) *P( y ) The addition rule therefore reduces to  P(x U y )=P( x ) + P( y ) - = P( x ) *P( y )

In both situations it will give the same output.

Hello students, Previously we have discussed about exponential growth and decay and in this section we are going to learn about the impossible event which comes under cbse 11th syllabus. Impossible events are those types of events that have zero probability of occurring. It can be shown like this: - Pr (A) = 0, where ‘A’ is an event. Zero means no size, magnitude, or quantity. We know that zero is neither positive nor negative while it is the additive identity. The simple means of impossible events is an event that will not happen.

Impossible events probability can be understand by the following example.

Example 1: - In a single toss of die, what is the probability of getting a number 8?

Solution: - We know that in tossing a coin, 8 will never come up.

So, getting 8 is an impossible event.

P (getting 8 in a single throw of a die) = 0 / 6 = 0.

Thus, in this example the probability of an impossible event is zero.

Impossible events are the reverse of the possible events.

The sum of the sure event's probability and impossible event's probability is 1. Where the probability of impossible events is zero and the probability of sure event is 1.

Example 2: - Suppose you have a bag of 10 balls then what is the probability of drawing a toffee?

Solution: - Probability (E) = 0 or P (E) = 0.

The bag only contains balls but it does not contain any type of toffee. So it is impossible to draw the toffees from the bag.

Probability (toffee) = 0 / 10 = 0,

Therefore, the probability of drawing a toffee is zero.

So at last it is specified that an event that has no chance of occurring is called the impossible events. Every event that has zero probability of occurring is known as impossible events.