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# Conditional Probability

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 Sub Topics Whenever we deal with conditional Probability we need to have knowledge about the word probability. Generally people have a rough idea about its meaning. In our day-to-day life we came across statement like, probably it may rain today, he may join politics or American team has a good chance of winning the world cup. When we use terms like: possible, chance, likely etc. all these terms convey the same sense that the event is not certain or in other words there is an uncertainty about the occurrence (or happening) of the event. Conditional probability consist of two events, these events can be independent or can be dependent. Suppose, if we have two events ‘A’ and ‘B’ then we have the conditional probability Formula: p(A/B)=p(B ∩ A)/p(A) and p(B/A) =p(A ∩ B)/p(A), p(A/B) is the probability of ‘A’ when ‘B’ has already occurred and p(B/A) is the probability of ‘B’ when ‘A’ has already occurred. With the help of conditional Probabilities we can find the probability of two Independent Events suppose if we are given a Set of playing cards and we are asked to find the probability of a black card greater than 4 and less than 5 than we can solve this problem with the help of conditional probability, let ‘A’ be the event for getting card greater than 4 and less than 5 and ‘B’ be the event of getting black card. Now we need to find the probability of ‘A’ when ‘B’ has already occurred and for that we have the formula given above with the help of this formula we can easily find the probability. This is all about conditional probability.

## Properties of Conditional Probability

In the statistical mathematics we study about the analytical processes to solve the problems. Conditional probability is most important concept of Statistics. Conditional probability is generally used for modeling the queries. Here we use the symbol (P) to denote the probability. More often P (X / Y) can be assigned intuitive consideration. By using the Conditional Probability we can compare quantities by computing other Probabilities. In the statistical analysis we perform the various tasks that are related to finding the result from the various observational processes. The conditional probability can be defined in symbols as:
P (X ∩ Y) = P (Y) P (X / Y)
The above given expression is a direct notation of the definition. The properties of conditional probability provide the process for analyzing the resultant outcome from the total events. In the probability we perform the estimation of outcome from the total generated events. Suppose John tossed a coin, which has the two faces that is head and tail. Then question arises that what is the probability for outcome coming head or tail.
In the general term if we solve the above question then we can say that the result will be head or tail but in the statistical mathematics the probability of outcome as head is ½ and probability of outcome as tail is ½. From the above solution we can say that probability can simply be calculated by performing the following formula:
Probability (P) = no of successful event / total Numbers of events.
Probability can be simply be expanded by using conditional probability. A conditional probability is the probability in which one event occurs only when the other event has occurred. In the case of die game when we throw the die at single time then probability will be 1/6 for any number (1 to 6) in the die. If we throw two dice then probability of an outcome on one die is also dependent on another die. The result generated from this situation is known as conditional probability. The conditional probability for an event ‘a’ given ‘b’ can be written as P (a / b).
In more standardized form conditional probability Formula is:
P (a/b) = P (a*b) / P (b)
In the more describable way we can say that event ‘a’ is independent of ‘b’, if the conditional probability of a given b is the same as the unconditional probability of a. Now here we describe the formula for the conditional probability properties. Suppose two events ‘a’ and ‘b’ are dependent on each other. Then the resultant probability will be (by applying conditional probability):
P (a and b) = P (a) * P (b / a)
This can be further solved as:
P (a and b) = P (a) * P (b / a)
ð P (a and b) / P (a) = P (a) * P (b / a) / P (a)
In above P (a) is canceled by the denominator value. So equation becomes:
P (a and b) / P (a) = P (b / a)
So, P (b / a) = P (a and b) / P (a)
The function P (X / Y) can be determined as a function of X. In this function, the condition is Y must be a fixed value. This condition is defining in the Kolmogorov axioms, which need to be satisfied. In the below rules are derived form the axioms in the statistical form.
a) Conditional probabilities with addition formula: In the conditional probability we use the concept of addition. Here addition operation for the Set shown by the (∪) symbol. Let’s show below:
P ( X1 ∪ x2 ∪ …....... ∪ Xn) = P ( X1 / Y ) + P ( X2 / Y ) + …......... for the events X1 , X2 up to Xn.
b) Complement rule for conditional probability: This operation provides the inverse of the given probabilities into the queries:
P (X’ / Y) = 1 – P (X / Y).

-auto;w�Js P 8" it-text-size-adjust: auto;-webkit-text-stroke-width: 0px;background-image: initial;background-attachment:initial;background-origin: initial;background-clip: initial; word-spacing:0px'>The property of a Random Variable say ‘Y’ is that if it is adjusted by multiplying by a value say ‘c’ and adding a value say ‘e’ then the Mean will become:
µe + cY = e + cµe; where ‘µ’ is the mean.
The main objective of this is to finding a unique or single value for a number of values or measurements in a given data set or sample. The related calculations of mean are Mode, arithmetic mean, geometrical mean, harmonic mean, root mean Square etc.
So this is all about the mean of discrete random variables.

## Conditional Probability Distribution

Conditional Probability is a mathematical probability that can be calculated depending on the fact that one event depends upon another. For example, you have been asked to find the probability of booking a parking ticket if you have to rush to a movie, or find that a response to an investigation query was "Yes," given that accused is your friend. Conditional Probabilities are generally questioned in form of certain queries found in statements; we will be writing it as P (A|B) in our mathematical calculations, which Mean "the probability of occurrence of event 'A', given event 'B'."
Suppose we have two events 'A' and 'B', such that they are mutually dependent on each other. Their probability of occurrence together can be given as:
P (A and B) = P (A) P (B)............... equation 1
So formula for Conditional Probability distribution can be derived using equation 1 as follows:
P (A and B) = P (A) P (B),
P (A and B) / P (A) = P (A) P (B) / P (A),
P (A and B) / P (A) = P (A) P (B I A) / P (A),
P (A and B) / P (A) = P (B I A),
Or P (B I A) = P (A and B) / P (A). This represents Bayes theorem Conditional Probability too.
Let us now consider some examples to understand conditional Probability Distribution:

Example :

A bucket contains yellow and white marbles. Two marbles are selected without adding more marbles to the bucket. The probability of selecting a yellow marble and then a white marble is 0.20, and the probability of selecting a yellow marble on the first draw is 0.40. What is the probability of selecting a white marble on the second draw, given that the first marble drawn was yellow?

Solution:

 P(White I Yellow) = P(Yellow and White) = 0.20 =0.50 = 50% P(Yellow) 0.40