Machine Learning - Probability and Counting Rules - Conditional Probability

In this session, we are discussing a very important topic that is conditional probability

Now, what is the conditional probability? When the outcome or occurrence of the first event affects the outcome or occurrence of the second event

in such way that the probability is changed. The events are said to be dependent events

Two events will be dependent if the probability of occurrence of the first event

affects the probability calculation of occurrence of the second event. If it is so, then you can say that

two events are dependent. The conditional probability of an event B in relationship to an event

A is the probability that event B occurs after event A has already occurred. So the notation

for the conditional probability is probability of BA, such that A has occurred

already. So, that means we are calculating the probability of occurrence of B after the

event A has occurred. So, when the event A has occurred, that will affect the calculation

of the probability of event B. So, probability of event B given event A. So when two events

are dependent, the probability of both occurring can be expressed in this way. That means probability

of A and B is equal to probability of A into probability of B given A So in this way you can write this one

So when two events are dependent, the probability of both occurring will become this one

So now let us go for one better understanding through some examples and all

But at first, we are considering this very example. There were five bugleries reported in 2003, 16 in 2004, 23, 232, in 2005

If a researcher wishes to select a random two bugleries to further investigation, find

the probability that both will be occurring on the same year, that is 2004

In this case, the events are dependent, and since the researcher wishes to investigate two

distinct cases, hence the first case is selected and not replaced. So now we are considering this one, so probability of C1 and C2 is equal to probability

of C1 into probability of C2 for given C1. So here we can write in this way, so 16 by 53, why 53

Because if you go on adding this respective 5 plus 16 plus 32, we are going to get here 53

So, 16 by 53, because we are interested to select two cases, buglery cases for the year 2004

So that why 2004 was having 16 number of bugleries So 16 by 53 into 15 by 52 So if you go on multiplying them we are going to get this respective value So this is known as this is known as the

conditional probability. So I think the previous slide is clear to us, and we have implemented

the same in this respective example. The probability that the second event B occurs given that

the first event A has occurred can be found by dividing the probability that both event occurred

by the probability that the first event has occurred. The formula can be written in this way

So, this is the respective formula whichever we are going to get finally. But let us derive

this formula. So from where you can derive? Because already we are having this one. This

formula we are already having. So, P of A and a. and b is equal to p of a into probability of b given a so we have written that one that is p of

a and b is equal to p of a into p of b given a now on the both side we are just dividing this one

using probability of occurrence of event a on the both side so now they will cancel out ultimately

we are remaining with the same with the value with this particular expression which we wrote here so in the

next problem we shall be applying this respective formula. So responses obtained on a survey question as below So male answers given yes and no total is this and female given the answers yes or no total is this So this number of total yes this number

of total no, and this is the total number of response that is 100. Find the probability that

the respondent has, was a male given that the respondent answered no. So let me repeat

the problem once again. Find the probability that the respondent was a male given that the

respondent answered no. So, the probability is to find probability of male given the answer

no in this way. We are writing this one. So, that is probability of male given the answer

no. So, now it can be written in this way. It is already we have written. That is probability

of p and a by probability of a. So, we have written the same thing. So, probability of

n and m by probability of n. So, what is probability of n and m at the same time, N

and m will be true? In that case, male and no. So, here we are getting this one as

18. So, 18 by 100. And probability of no, here the sum is 60. So 60 by 100. If you go

on doing the simplification, the probability does obtain is 3 by 10. So, in this way

we have explained with some examples that how this conditional probability can be calculated

for a given problem. Thanks for watching this video