 Hello and welcome to the session. My name is Mansi and I am going to help you with the following question. The question says three coins are tossed once. Let A denote the event three heads show, B denote the event two heads and one tail show, C denote the event three tails show and D denote the event a head shows on the first coin. Now which events are mutually exclusive, simple and compound. That means first we have to tell which events are mutually exclusive then which events are simple and which are compound. So let us start with a solution to this question. Now first of all let us write down the sample space of the event that three coins are tossed. So the sample space will be H H H that means we can get heads in three coins, H H T that means we may get head in first two coins and tails in the third coin or H T H, H T T, T H H, T H T, T T H and T T T. So this is a sample space when three coins are tossed. Now let us write down the sample space for the event A. Event A was when three heads show. So the sample space will contain just one sample point that is H H H. Now we write down the sample space for event B. Event was two heads and one tails show. So all such sample points are H H T, H T H and T H H because here two heads and one tails show in each sample point. Now event C was three tails show. So that will be T T T will be the only sample point and event D was a head shows on the first coin. So the sample points would be H H H, H H T, H T H, H T T. Now we see that in the event D we have a head shows on the first coin. So no matter what we have on other coins we consider all those sample points where we have head on the first coin. Since here we have getting three heads on three different coins and so on. So here also we have the same thing getting heads in the first coin, heads in the second coin and tails on the third coin and so on. Now we have to tell which events are mutually exclusive. So we see that for two events say event E and event F to be mutually exclusive we should have E intersection F should be equal to Phi that means the sets E and F they should have no points in common. So that means these sample space E and F should not have any sample point in common. Then we say that E and F are mutually exclusive. So by this definition we can say that we have to find A intersection B we see that there is no point common in A and B. So A intersection B is Phi. Similarly A intersection C is Phi. Now we see that A intersection D is the sample point H H H. So A intersection D is the sample space of H H H that is not equal to Phi. Now we find out B intersection C that is also equal to Phi because B and C they have no sample points in common. Now B intersection D is the sample space with sample points H H T and H T H that is not equal to Phi and we see C intersection D since C and D they do not have any points in common. So C intersection D is also equal to Phi. So we can say that answer to first part of the question is A and B A and C B and C C and D they are all mutually exclusive because A intersection B is Phi A intersection C is Phi B intersection C is Phi and C intersection D is Phi. Now we see that if an event E has only one sample point of a sample space it is called a sample event. So by this definition we can say that since A and C they have just one sample point of the sample space so they are simple events. So our answer to second part of the question is A and C and we also see that if an event has more than one sample point it is called a compound event. So from this we can see that event B and event D they have more than one sample point so they are compound events. So our answer to third part of the question is B and D are compound events. So this is our answer to the question. I hope that you understood the question and enjoyed the session. Have a good day.