 Dear students, I'm now going to present to you the concept of an event in the case of a two-dimensional space. Yanni, in that case, when we have a set of ordered pairs. So let us begin by saying that capital D is the space associated with the random vector, X1, X2. And let capital A be a subset of D. And just as we have in the case of one single random variable, we know that in the case of a single random variable, an event is exactly this, that it is a subset of the sample space. So, if A is a subset of D consisting of ordered pairs, something, comma, something, or usra ke kai pairs, then we can say that A is an event in this two-dimensional space. And then, once we've done that, after that we can attempt to compute the probability of that particular event. So let us try to understand all this with reference to an example. Suppose that a coin, a fair coin, is tossed three times and our interest is in the ordered pair X1, number of heads on the first two tosses, comma, X2, number of heads on all three tosses. Then, first of all, what is the sample space? Art, uskender triplets, ke li jay. Tail, tail, tail, tail, tail, head, tail, head, tail, tail. Tail, head, head, head, tail, head, head, head, head. Ye to art triplets, innit? But that is the sample space of our tossing process. And then, if X1 is the number of heads on the first two tosses and X2, the number of heads on all three tosses, then, for example, X1 for the outcome, head, tail, head, will be equal to one. Isli ye ke pehle do ke andar, head, tail ke andar, sirf ek head hai. Or X2, for this very same outcome, head, tail, head, that will be two, ke all three ko agar hum ekatha dekhayin, to there are two heads. So, X1, X2, ye jay ordered pair hai, iski jay value ai hai for this particular situation that is one comma two. Continuing in this way, we can define the space D, consisting of all possible ordered pairs that we could have got according to the definition that I have just now presented. So, you can do that yourself, and I'm just going to present that entire space to you now. So, D is equal to zero comma zero dekhayin. Agar aap tail, tail, tail ko dekhayin, to pehle do ke andar bhi koi head nahi hai, aur tino ke andar bhi koi head nahi hai. So, zero, zero is the very first possible value of the random vector, may I say, X1 comma X2. Iske baad aga, kya ho ga? Zero, one. Ab ye kab ho ga? Dekhayin. Tail, tail, head. Pehle do ke andar tail, tail me koi head nahi hai, to zero aga. Aur tino ko ekatcha karke dekhayin, to zero, tail, tail, head. So, there is one head. So, comma ke baad, one, zero, one. Istra chalte jayin, and you have all of them. Now, ye to hamari space aagayi, the two, consisting of these ordered pairs, so these are the two-dimensional jo ho gaya nahi, X1 comma X2, iske jo all possible values thi under this situation, jo hamari definition of X1 or X2 ki, ko hamare saamne aajati. Ab issare ke baad, we are in a position to define various events, and let me take one. Suppose that I'm interested in the event that the number of heads on the first two tosses is greater than zero, and the number of heads on all three tosses is less than two. All right? Me is event me interested in ke jee, pehle do jo ham, usme jo number of heads aaye vo zero se jada ho. To ab bata aaye vo korn korn se ho sakte hain. Zahire ke vo one head bhi ho sakta hain pehle do tosson ke andar, mere event ke mutabik. Or do no heads bhi ho sakte hain. So, zero se to zyada hi hain aajee. Magar saati me ne kya kaayi ki ye ho, and number of heads on all three tosses should be less than two. So agar all three ke andar less than two ho gaa, to kya ho sakta hain? Agar tail, tail, tail ho, to zero hain, to less than two hain. Or agar tail, tail, head ho, yaa, tail, head, tail ho, yaa, head, tail, tail ho, to kitne aah heads hain? One, to two se kaam hain na? To ab goya aapne samaj liya ke vo pehli wali kondition ko poora karne ke liye, we can have before the comma, either one or two, isliye ke greater than zero chahiye. Or jo after the comma hain, vo aapne samjha ke zero yaa one, yuke less than two chahiye. Lekin my dear students, this is not the case. You see because we also have to apply common sense, this particular event that I have defined, X1 greater than zero and, ye jo and hain na? This word is of primary importance here. Vo greater than zero ho, or and vo dosara jo hain vo less than two ho, this can only happen if we have one comma one. Dhekein, agar pehli do ke andar one aaje hain aur baad me aur na aayi to phir vani rahega na? To ab ye dono konditions bayaq vaakth simultaneously fulfil ho rahe hain, jaehsa ke ho nahi chahiye. One one agar hain to X1 is greater than zero and X2 is less than two. Dhekein vo jo agar aap kahin ke ji two one bhi ho sakta hain, to is that all right? Two one ka matlab kya hoa? Ke ji pehli do dafa me two aage hain, hain heads to vo zero ho se zada hain to theek hain, lekin theek nahi hain. Kyuke agar pehli dafa me two aage hain, to phir agli dafa nahi bhi aaye to two to aage hain hain. To ho to humari to konditions thi ke wo teenok me mila aake they have to be less than two. Two nahi, less than two. So this is how we can see that the event a defined as I have defined is the one loan ordered pair one one. And coming back to that previous point ke event jo hota hain chahe wo univariate situation hain, chahe wo bivariate situation hain jasa ke yeh hain, or even if it is a multivariate situation, the event would always be a subset of the big set which would contain all possible outcomes pertaining to that particular experiment.