 Hello and welcome to the session. In this session we will discuss probability. Let s be a sample space containing outcomes w1, w2 and so on up to wn the number pwi associated with sample point w i such that pwi is greater than equal to 0 and less than equal to 1. For each w i belongs to s then pw1 plus pw2 and so on up to pwn is equal to 1. Next is for any event a p of a is equal to summation pwi w i belongs to a. This number pwi is called the probability of the outcome w i. Consider an experiment of tossing a coin in that case the sample space s would be equal to ht. Now probability of getting a head would be equal to 1 upon 2 and the probability of getting a tail will also be equal to 1 upon 2. Clearly you can see that each number is neither less than 0 nor greater than 1. That is each probability lies between 0 and 1. And sum of the probabilities that is probability of head plus probability of the tail is equal to 1. All outcomes with equal probability are called equally likely outcomes. Next we consider probabilities of equally likely outcomes. For a sample space with equally likely outcomes probability of an event given by a is equal to na upon ns where this na is the number of elements in set a and ns is the number of elements in set s. Or we can also say probability of an event a is equal to number of outcomes favorable to a upon total possible outcomes. Suppose the two coins are tossed once so the sample space s would be given by hh ht th tt so ns that is number of elements of s is equal to 4 or you can say total possible outcomes is 4. If we consider e1 to be an event of getting two heads then e1 would be equal to hh then ne1 would be equal to 1 that is number of elements in the set e1 is 1 or you can say number of outcomes favorable to the event e1 is 1. So probability of getting two heads that is probability of the event e1 is equal to number of elements in the set e1 that is 1 upon number of elements in the sample space s that is 4. So 1 upon 4 is the probability of getting two heads. Next we discuss probability of the event a or b. If we have two events a and b then probability of the event a or b which is given by probability of a union b is equal to probability of the event a plus probability of the event b minus probability of the event a and b that is probability of the event a intersection b. If the events a and b are mutually exclusive events then we have a intersection b would be equal to phi so probability of a intersection b is equal to 0. Hence in that case probability of a or b that is a union b would be equal to probability of a plus probability of b. In the same example that we had considered above of tossing two coins where we have taken even to be the event of getting two heads let's consider e2 to be the event of getting at least one head. Now in that case e2 would be equal to ht th hh that is we have ne2 that is number of elements in the set e2 is equal to 3. So now probability of getting at least one head that is probability of the event e2 would be equal to 3 upon 4. Now even intersection e2 is equal to hh then n of even intersection e2 is equal to 1 and so probability of even intersection e2 that is probability of getting two heads and at least one head is equal to 1 upon 4. Now probability of getting two heads or at least one head that is probability of even union e2 is equal to probability of even plus probability of e2 minus probability of even intersection e2. So this becomes equal to probability of even that is 1 upon 4 plus 3 upon 4 that is probability of e2 minus probability of even intersection e2 that is 1 upon 4 that comes out to be equal to 3 upon 4. So we have probability of getting two heads or at least one head is 3 upon 4. Next we have probability of event not a. If we consider a to be any event then probability of not a or you can say probability of a complement would be equal to 1 minus probability of a. In the experiment of tossing two coins this is the sample space as even is the event of getting two heads and probability of getting two heads that is probability of even is 1 upon 4. Now probability of not even that is probability of not getting two heads or you can say probability of even complement would be equal to 1 minus probability of even that is 1 minus 1 upon 4 and that is equal to 3 upon 4 that is probability of not even probability of event complement. This complete session hope you have understood how we find the probability of an event.