 Hello and welcome to the session. In this session, first we will discuss random experiments. In our day-to-day life, people perform many activities which have a fixed result no matter any number of times they are repeated. Like when we toss a coin, it may turn up a head or a tail but we are not sure which one of these results will actually be obtained. Such experiments are called random experiments. So we define an experiment is called random experiment if it satisfies the following two conditions which says it has more than one possible outcome then next one is it is not possible to predict the outcome in advance. So when these two conditions are satisfied in an experiment that experiment is called the random experiment. Like for example tossing a fair coin is a random experiment because in this case we have more than one possible outcome that is when we toss a coin we either get head or a tail and in this case we cannot predict the outcome in advance also. So tossing a fair coin is a random experiment. Next we have a possible result of a random experiment is called its outcome. Like for example in the experiment of rolling a die the outcomes would be 1, 2, 3, 4, 5, 6 and the set of all possible outcomes is called the sample space it is denoted by the letter s in the experiment of rolling a die the sample space s would be the set of all possible outcomes and the possible outcomes in this case is this so we have s is equal to 1, 2, 3, 4, 5, 6 this is the sample space of the experiment of rolling a die. Then the elements of the sample space are called sample points. We can also say that each outcome of the random experiment is called the sample point like in the experiment of rolling a die these outcomes are the sample points 1, 2, 3, 4, 5, 6. Next we discuss event any subset e of a sample space s is called an event. Like in the random experiment of rolling a die this is the sample space given by s consider event of getting a prime number in this case this would be given by e equal to the set 2, 3, 5 as you can easily observe that this event e is the subset of the sample space s. Next we have occurrence of an event consider w to be the outcome of the experiment if the outcome w of the experiment is such that w belongs to the event e then we say the event e of sample space s has occurred and if in case we have the outcome w does not belong to the event e then event e of sample space s has not occurred. Next we discuss types of events. Events can be classified into various types on the basis of the elements they have. First we have the kind impossible event basically the empty set phi are the sample space s describe events this set phi that is the empty set phi is called an impossible event. Like in a throw of a die the sample space s is given by 1, 2, 3, 4, 5, 6 if we consider event e1 be an event of getting a number less than 1 as you can clearly see that no outcome can be less than 1 so this event e1 is considered as impossible event. Next we have sure event the whole sample space is called the sure event like in the same experiment of throwing a die if we consider e2 to be an event of getting a number less than 7 as you can see each outcome is a number less than 7 so we can say that e2 is a sure event. Next is the simple event which says if an event e has only one sample point of a sample space it is called a simple event or we can also say elementary event like in an experiment of tossing two coins the sample space s would be given by h h h t h t t let's consider event to be an event of getting a tail on both the coins then even would be equal to t t so this event e1 is a simple event since it has only one sample point of the sample space s next we have compound event that is if an event has more than one sample point it is called a compound event like in the same example of tossing two coins if we take e2 to be an event of getting at least one tail then this e2 would be equal to h t h t t this event e2 is the compound event since it has more than one sample point next we discuss algebra of events first we have complementary event for every event a there corresponds another event a dash which is called complementary event to a this is also called event not a we can also say that the complementary event a dash is equal to the sample space s minus the event a clearly a dash occurs when a does not occur and a occurs when a dash does not occur so in a trial one of the two events a or a dash is sure to occur like in the experiment of tossing two coins this is the sample space s and if even is the event of getting a tail on both the coins then even is equal to pt even dash would be equal to that is the complementary event of even would be equal to s that is the sample space minus even and that is equal to h h h t th so this is the complementary event to even next we have event a or b this event a or b is given by a union b that is a union b is the event either a or b or both which could also be given by omega such that omega belongs to a or omega belongs to b next we have event a and b event a and b is given by a intersection b this could also be written as omega such that omega belongs to a and omega belongs to b next is event a but not b this is given by the set a minus b which could also be written as a intersection b complement that is this is the set of all the events which are in a but not in b next we have mutually exclusive events events a and b are set to be mutually exclusive if we have a intersection b is equal to 5 in general we say that the a and b are mutually exclusive events if they cannot occur simultaneously in that case the sets a and b are disjoint simple events of a sample space are always mutually exclusive in an experiment of throwing the dye this is the sample space consider even to be an event of getting a number less than 3 then even would be equal to 1 2 then consider e2 to be an event of getting a number more than 4 in that case e2 would be equal to 5 6 clearly even intersection e2 is equal to 5 so we say that even e2 are mutually exclusive events next we have exhaustive events if we have events even e2 and so on up to en such that we have even union e2 union e3 union so on up to union e n that is equal to union i goes from 1 to n e i is equal to the sample space s then in that case the evens even e2 and so on up to e n are called exhaustive events or we can also say that the evens even e2 and so on up to e n are set to be exhaustive if at least one of them necessarily occurs whenever the experiment is performed if we have e i intersection e j is equal to 5 when i is not equal to j and union e i i goes from 1 to n is equal to the sample space s then in that case the evens even e2 and so on up to e n are called mutually exclusive and exhaustive events in an experiment of rolling a dye this is the sample space even is the event of getting a number less than 3 e2 is the event of getting a number more than 4 consider e3 to be an event of getting number more than 2 and less than 5 then e3 would be equal to 3 4 now as you can see even intersection e2 intersection e3 is equal to 5 and even union e2 union e3 is equal to the sample space s so we say that the evens even e2 e3 are mutually exclusive and exhaustive events this completes the session hope you have understood the basic concepts like what is an event different types of events and the algebra of events