Hello everyone, welcome to the course on machine learning with Python. In this video, we will learn a little bit about introductory probability theory. Now, why probability theory is important in studying machine learning. Machine learning is all about finding the patterns exhibited by the data. Machine learning tells that the observations are noisy instances of math simpler room to learn the truth from the data is the job of the machine learning algorithm. As the rules are the patterns exhibited by the data are not explicit.

Hence, there is some degree of uncertainty associated with the learned rule from the data. Now we know that probability theory is a mathematical framework to deal with the uncertainties. Hence, probability theory is the indispensable tool to study we should know what is probability probability is the mathematical description of randomness and uncertainty. It is a way to measure or quantify uncertainty. Another way to think about probability is that it is the official name for charge Now probability is the likelihood of something happening. One way to think of probability is that it is the likelihood that something will occur.

Probability is used to answer the following types of questions. Let's say what is the chance that it will rain tomorrow? Or what is the chance that a stock price will go up in tomorrow? What is the probability that I will win the lottery, so each of those examples has some degree of uncertainty for some, the chances are quite good, so the probability would be quite high. For others, the chances are not very good, so the probability is quite low. Now few notions a random experiment is a process whose all possible outcomes are known to us but which outcome will occur in a particular trial is unknown.

Until the experiment is performed. sample space of an unknown experiment is the collection of all possible outcomes of that experiment and event is the subset of the sample space, which describes some possible outcomes of a random experiment. For example, rolling a dice is a random experiment whose sample spaces is 123456. All the six outcomes and an even could be obtaining an even number as outcome. Few notations if capital A, we had even then P of A, which is written as p within parentheses a denotes the probability of the event, the probability of an event tells how likely it is that the event will occur. Now, probability is relative frequency to estimate the probability of an event a written as p of a we may repeat the random experiment many times and count the number of times the event e occurs then probability of A is estimated by the ratio of the number of times a occurs to the number of repetitions of the experiment which is called the relative frequency of the event A.

So, the relative frequency of event A is basically the issue of number of times that even he has occurred to the total number of reputations Experience no law of large number tells that the actual or the true probability of an event A is estimated by the relative frequency with which the event occurs in a long series of trials. So, when an ordinary fear die is rolled once what is the probability that the number of rolled is even will denote the event by capital E for Even so, we will introduce it to find probability of E let's analyze the problem first. So, the random experiment is rolling a fair die once the sample space of all possible outcome in this case is noted by a set 123456. Since the die is fair, this means all the six possible outcomes are equally likely each having probability of one by six of awkward we are interested in a particular type of outcome which is represented by event capital E getting an even number since three out of six equally likely outcomes make up the event that is outcome.

246 Has probably of he would be three up and six which is point five. Now basic rules of probability, the probability of any event A will be between zero to one, okay. So this is what is called the range rule. So the maximum value that a probability of event A or any event is one, and the minimum value is zero. So probability is always a proper fraction, probability of the sample space is one, the sum of all probabilities of all possible outcomes is one probability of a compliment where E is an event is given by one minus the probability of the event A here e complement is nothing but the sample space minus the event probability of A union B is equals to probability of A plus probability of b minus probability of A intersection B. This is also called the addition rule of probability.

Now if A and B are disjoint event that means there is no common co occurrence of the events then A intersection B will be null then Probability of A union v would be probability of A plus probability of B, probability of A intersection B is equals to probability of A multiplied with probability of B provided that A and B are independent event that means one does not depend on another or occurring of one event does not supply any additional information about occurring of another event, probability of A union B union C is equals to probability of A plus probability of B plus probability of c minus probability of A intersection v minus probability of B intersection C minus probability of A intersection C plus probability of A intersection B intersection C. Now what if E and B are not independent event? Okay, so let's see something called conditional probability.

So probability of A given B, so how it is actually written, ie with a slash or horizontal bar. B is called the probability of A given B is equals to probability of A intersection B divided by probability of B okay for example, consider the following table which describes the smoking habit of few persons, no gender male, female and this is smoker and nonsmoker. So, there are 187 smelled smoker 43 will not smoker and there are 57 female smoker and 203 female nonsmoker okay. There are total 240 number of males and 260 number of females there are total 244 number of smoker and 256 number of nonsmoker and there are 500% or here actually get positive student data. So total 500 students are there. Now let in the notes the effect of being mean why capital V represents the event of being smoker?

What is the conditional probability that a randomly chosen student is a smoker even that the student is female? Okay, so we know that the student is female, what is the probability that This randomly chosen person is smoker okay. So, we want to find out probability of smoker given female. So, that is nothing but probability of smoker and female divided by probability of female. So, what is probability smoker and female? So, how many smoker and female out there total 57 and how many Protestants there 500 So, probability of smoker and female is 57 by 500 whereas, how many female students at their current 260 So, the probability of having a female student is 260 upon 500.

So, the when we simplify this it becomes 67 by 260 it is nothing but 0.2192. Okay, so that is how conditional unity is calculated. Now, let's move forward, which is called the multiplication rule of probability. So, the condition code tells us that probability of A given B is equals to probability of A intersection B divided by probability of B now rerun this equation we can write probability of A intersection B is nothing but probability of b multiplied with probability of A given B is akin probability of A intersection B is nothing but probability of A multiplied with probability of B given A. Why, because probability of b given a is nothing but probability of A intersection B divided by probability of A hence we can write that probability of A intersection B is nothing but probability of b multiplied with probability of A given B or it is same as probability of A multiplied with a to be given a, this is known as multiplication rule of probability now lit.

Problem A and B are not dependent on each other. That means a recording of one event does not supply any additional information on the probability of offering another event. In that case, probability of A given B would be nothing but probability of A and probability of B given A will be nothing but 80 of B in that case, when a and b are independent of each other probability of A intersection B would be nothing but probability of A multiplied with probability of B, we have discussed this rule already a law of total probability for the total probability rule, also called the law of total probability breaks up the probability calculation into distinct parts. It is used to find the probability of an event A when you do not know enough information about a. So instead you take a related event B and use the probability of B to calculate the probability of A so to speak mathematically, let's say this is the total sample space and this is B even B and this is B complement this yellow.

Now let's say this is the event A. So A intersection B is this portion, and A intersection B complement is this portion. So does the total probability rule in this case would be probability of A is nothing but probability of A intersection B plus probability of A intersection B complement. Now for multiplication rule of probability, we can rephrase this as probability of A is nothing probability of B multiply with probability of A given B plus probability of B complement multiply the probability of A given B complex. Okay, let's see an example 80% of the people attend their primary care physician regularly. 30% of these people have no health problems proper during the following year.

Out of the 20% people who don't see the doctor regularly, only 5% have no health issues during the following year. What is the probability a random person will have no health problems in the following year so let's hear the event of person having no health problem is denoted by a and people seeing doctor is noted by P then by theory of total probability probability of A would be nothing but only to have A intersection B plus probability of A intersection B complement then by multiplication rule of probability we can always expand this adds probability of as opposed to probability of b multiplied by the quantity of A intersection B does To be comfortable and multiply with probability of A intersection B complement, so probability of A is equals 2.8 multiplied with point three, five, and one minus point eight multiplied with 2.05. So this is calculated to be 0.29.

Now for multiple events, this can be extended as probability of ease equals to summation over items from one to N probability of A intersection B suffix. And using multiplication rule a probability we can extend the probability of A Okay, now let's go to Bayes rule. for multiplication probability. We know that probability of B multiple probability of A given B is nothing but probability of A multiplied the probability of B given A so we can always rearrange this and write these as probability of b given a is equals to probability of A given B divided by probability of A multiplied with probability of B assuming that probability of a party goes to zero, okay, so this is what is called the Bayes rule. In the next video, we will learn about About exploratory data analysis, okay. And I will suggest you to go through any eliminated probability book to learn more about probability.

See you in the next lecture. Thank you.