Conditional Probability Rule

Definition:

The conditional probability if an event B,assuming that the event A has already happened; is denoted by P(B/A) and defined as    P(B/A)=P(A nn B) / P(A)   ,  provided P(A) != 0.

similarly,  P(A/B)=P(A nn B) / P(A)   ,  provided P(B) != 0 .

Explanation :

For instance consider the following example to understand the concept of conditional probability.

Suppose a fair die is rolled once. The sample space is S = { 1,2,3,4,5,6 } .

Now try to answer these questions :

Q 1: What is the probability that getting an even number which is less than 4  ?

Q 2: If the die shows an even number, then what is the probability that it is less than 4 ?

case 1:  The event of getting an even number which is less than 4 is {2}

Therefore , P1 =n({2}) / n({1,2,3,4,5,6})= 1/6

case 2:  Here first we restrict our sample space S to a subset containing only even number i.e. to {2,4,6}. Then our interest is to find the probability if the event getting a number less than 4 i.e. to {2}.

Therefore,  P2 = n({2}) / n({2,4,6})  = 1/3

In the above two cases the favourable events are the same, but the number of exhaustive outcomes are different. In case 2, we observe that we have imposed a condition on sample space, then asked to find the probability.This type of probability . This type of probability is called conditional probability.

Explanation:

A conditional probability is the probability of an event given that another event has occurred. For example, what is the probability that the total of two dice will be greater than 8 given that the first die is a 6? This can be computed by considering only outcomes for which the first die is a 6. Then, determine the proportion of these outcomes that total more than 8. All the possible outcomes for two dice are shown below:

all outcomes two dice

There are 6 outcomes for which the first die is a 6, and of these, there are four that total more than 8 (6,3; 6,4; 6,5; 6,6). The probability of a total greater than 8 given that the first die is 6 is therefore 4/6 = 2/3.

More formally, this probability can be written as:

p(total>8 | Die 1 = 6) = 2/3.

In this equation, the expression to the left of the vertical bar represents the event and the expression to the right of the vertical bar represents the condition. Thus it would be read as "The probability that the total is greater than 8 given that Die 1 is 6 is 2/3." In more abstract form, p(A|B) is the probability of event A given that event B occurred

conditional probability :


A clearcut description of Conditional Probability

The conditional probability if an event B,assuming that the event A has alteady happened; is dinoted by P(B/A) and defined as    P(B/A)=P(A nn B) / P(A)   ,  provided P(A) != 0.   A conditional probability is the probability of an event given that another event has occurred.