Joint Probability

Author: nitin.p070

Probability in math is defined as the chance of happening something in future. The two random variables A and B are defined on the same probability space, the joint probability distribution for A and B defines the probability of events defined in terms of both A and B. In the case of having only two random variables, this is called a bi-variate distribution, but the thought simplify to any number of random variables, giving a multivariate distribution.

Joint probability:

A numerical measure where the chance of two events happening together and at the same time are calculated. For example if the probability of event B happening at the same time event A happens, then the Joint probability has been given as follows.
Joint probability notation takes the form:

P (A `nn` B) or P (A and B)

Indicates the joint probability of A and B.

Example: The joint probability can be calculated by rolling a 2 and a 5 with two dissimilar dice.

with and without replacemant-Joint probability:

Joint probability is used in multistage testing .Joint probability can be done with replacement or without replacement.

With replacement: It indicates that the thing that are chosen on one stage are returned to the sample space before the next choice is made .For example, tossing a head on the first toss does not affect the outcome of flipping the coin a second time.

The probability that independent events A and B occur at the same time can be found by using the multiplication rule, or the product of the entity probabilities.

Example 1:

If you pick two cards from the deck without replacement, find the probability that they will both be aces.

Solution:

Total number of aces in the deck of cards = 4.

Cards picked up = 2 aces.

total number of aces* (total number of aces-1)
Hence the probability = --------------------------------------------------
total number of cards*( total number of cards-1)

P (AA) = `(4/52)*(3/51)` = `1/221` .

Without replacement:

Uses the same idea, if the first choice is not replaced only we consider the change in the sample space. Still we use the multiplication rule, but for each of the stages the numerator and/or denominator decreases

Example 2:

Find the probability of tossing a fair coin twice in a row, getting heads both times.

Solution:

While tossing a fair coin once we get head or tail.

Given that while tossing a coin head occurs,So

We know that the probability =(Number of favourable outcomes/Total number of outcomes)

Therefore probability of getting head while tossing the coin once P(H)= `1/2.`

Similarly tossing a coin next time we have probability of getting head P(H) =`1/2.`

As the question is to find the probability of tossing a fair coin twice in a row, getting heads both times it indicates that we have to find the joint probability without replacement.

As the probability of tossing a head is ` 1/2 ` each time P (H,H) =` (1/2) *(1/2) = 1/4.`

Article Source: http://www.articlesbase.com/science-articles/joint-probability-6618060.html

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Difference Of Sets

Author: Math Help

Here we are going to learn about an operatation on set called difference of sets.In mathematics, A set can have a limited  number of elements.Set is a collection of data.We can perform many operations on set.The difference operation is one of them.The subtract(difference) symbol in the  function represents the removal of the values from the second set from the first set. The operation of subtraction  is a removing or taking away objects from group of object.


Explanation of difference of two sets

Difference of sets is  defined as a method of rearrange  sets by removing the elements which belong to another sets. Difference of sets is a denoted  by either of the symbols – or \. P minus Q is can be written either P – Q or    P \ Q.

The differences  of two sets P and Q, is written as P-Q.it contains  contains that  elements of P which are not present  in elements of Q. here result  P-Q is obtained take set P as usual and compare with set Q. now remove those element in set P which matches with set Q.  If P={a,b,c,d}and Q={d,e}, then P-Q={a,b,c}.


Definition for difference of sets

The difference between the two sets A and B represented in the order as the set of all those elements of A which are not in B. It is denoted by A-B.

In symbol we write it as

A-B = {x :`x in A` and `x !in B`   }

similarly  B-A =   {  x  :   `x in B` and `x!in A`  }

By representing it in the venn diagram,


Examples problems

Below are the problems based on difference of sets -

Problem 1:

Consider the two sets A = {11,12, 13, 14, 15, 16}, B = {12, 14, 16, 18} find the difference between the  two sets?

Solution:

Given A = {11,12, 13, 14, 15, 16}

B = {12, 14, 16, 18}

A –B = {11, 13, 15}

B –A = {18}

The set of all elements are present in A or in B. But not in both is called the symmetric difference set.


Problem 2:

A={2,3,4,1,8,9}   B={2,3,4,1,8,,12},What is the A-B and B-A?

Solution:

Given A={2,3,4,1,8,9}

B={2,3,4,1,8,12}

Here all elements of A  is available in B except 9.

So the difference A-B ={ 9 }.

Here all elements of B  an available in A except 12.

So the difference B-A = {12 }.



Example 3:

Consider two sets A={a,b,f,g,h}, B={f,g,a,k} find A-B and B-A?

Solution:

Given  A = { a , b , f , g , h }

B = { f , g , a , k }

so   A-B  = { b , h }

and  B-A = { k }


Problem 4:

Consider given sets P={19,38,57,76,95} Q={7,19,57,75,94} Find P-Q and Q-P

Solution:

Given P = {19,38,57,76,95}

Q = {7,19,57,75,94}

so P-Q = {38,76,95}

and Q-P = {7,75,94}

Article Source: http://www.articlesbase.com/k-12-education-articles/difference-of-sets-6619193.html

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Probability Survey

Author: nitin.p070

The probability survey is the way of expressing an event that will occur. The probability survey is the event, the experiments that are repeatedly done under some predefined conditions. The results for one or more experiments are not equal. These types of experiments are called as the random experiments or simply experiments. The probability includes the sample space, trail and different forms of events.

Terms present in the probability survey:

Sample space indicates the total number of possibilities for an experiment.
Trial corresponds to the experiment is performed.
Event specifies the outcome of the experiments.
Exhaustive events are an event which contains all the necessary possible outcomes of the experiment.
Mutually exclusive events are the two events that cannot occur simultaneously.
The probability certain likely defines the equally likely event in the probability. Equally likely event means that the two or more events have an equal probability. For example while tossing the die the probability for getting the tail and also the probability for getting the head are the equally likely events. The equally likely event determines the equal probability for the events.

Example problems for probability survey:

Ex 1 :A jar has 6 gray and 9 red marbles. What is the probability to get one gray marbles from the urn without replacement?

Sol:

The number of marbles in the jar is 6 gray and 9 red marbles.

The total numbers of marbles are 15 marbles.

The possibility for getting a gray ball is 6.

The required probability is 6/15 .

Ex 2 : While tossing a fair die, find the complementary probability of the numbers greater than 3.

Sol:

The sample space for the die is S= {1, 2, 3, 4, 5, 6}

The total number of sample space =6.

A is the event for getting the number greater than 3.

A= {4, 5, 6}

The number of events greater than 3 is n (A) =3

P (A) =n (A)/ n(S)

P (A) = 3/6

P (A) = 1/2

The probability for getting the numbers greater than 3 is 1/2 .

The formula for the complementary probability is 1- P (original probability).

The required probability = 1-P (A)

The required probability = 1- 1/2

The required probability = 1/2

The complementary probability for the numbers greater than 3 is 1/2 .

Survey of probability of certain likely events:

Some examples for probability certain likely:

Probability for getting the head and the tail when a coin is tossed only one time.
The probability for getting the number 3 and number 4 are equally likely events.
If an urn contains 5 white balls and 5 red balls. In that the probability for getting the single white ball and also the probability for getting the single red ball are the equally likely events.

Ex 3 : A jar has 5 gray and 7 green marbles. What is the probability to get one gray marbles and also probability for getting 1 green marbles? Determine whether the above events are equally likely events.

Sol:

The number of marbles in the jar is 5 gray and 7 green marbles.

The total numbers of marbles are 12 marbles.

The possibility for getting a gray marble is 5.

The probability for getting one gray marble is 5/12.

The possibility for getting a green marble is 7.

The probability for getting one green marble is 7/12.

The probabilities are 5/12 and also 7/12. These two probabilities are not the equally likely event because the probability of that two events are not same they are different.

Ex 4 : A single six face die is rolled. Find the probability for getting the number 6 and also 3. Determine whether these two events are equally likely events are not.

Sol:

The sample space for the die is S= {1, 2, 3, 4, 5, 6}

The total number of sample space is 6.

The probability for getting the number 3 is 1/6 .

The probability for getting the number 6 is 1/6 .

The probabilities for the two events are 1/6 and 1/6 respectively. The probabilities for the two events are equal. So these two events are equally likely events.

Practice problems:

Two coins are tossed at the same time. What is the probability to get two tails?
Ans: 1/2 .

Article Source: http://www.articlesbase.com/k-12-education-articles/probability-survey-6618083.html

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