 Hello and welcome to the session. Let us discuss the following question. It says, A fair coin is tossed four times and a person win rupee 1 for each head and lose 1 rupee and 50 paisa for each tail that turns up. From this sample space calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts. Let us now move on to the solution. Let us first write the sample space S for the experiment of tossing a coin four times. Now since we are tossing a coin four times so the total number of outcomes will be equal to 2 to the power 4 since when we toss a coin we have two possibilities either we get a head or we get a tail. So the number of outcomes becomes 2 to the power 4 which is equal to 16. Let us now write the sample space. One of the possibility is that we get all the head. Then the second one is that we get head in first three tosses and then we get a tail or the other one is we get head into first two tosses and then we get a tail and then again we get a head. Then we get a head then we get a tail then again we get two heads. And after that may possible we get tail in the first toss and then we get three heads. Similarly we write all the possible outcomes. So this is the sample space of the experiment of tossing a coin four times. Now in all the outcomes we see that either we get all the heads or we get all the tails or we get head three times and tail one time or we get two heads and two tails or we get three tails and one tail. So there are five possibilities and we are asked to find the different amounts of money we can have after four tosses. So since there are five cases we will have five different amounts of money after four tosses. So the first case is if all head appears. We are given that person win rupee one for each head. So if all head appears then the person wins rupees four after four tosses he win rupees four. So this is one of the possibility that he wins rupees four. Now the second possibility is that the person gets three head one tail. Three head appears and one tail appears then the person wins rupees three since he gets three heads. So he wins rupees three minus the amount which he lose which is rupees one point five zero that is one rupee fifty paisa. So this is equal to one rupee fifty paisa so the person wins rupees one point five zero. Now the third possibility is if two head two tails appears. If two tails appears then the person loses rupees three minus the amount which he wins. He gets two heads so he wins rupees two so the person loses rupees three minus two that is rupee one. Now the next possibility is that the person get one head and three tails. So the person loses rupees he gets three heads then he lose three into one point five zero minus the amount which he wins which is rupees one. So he loses three rupees fifty paisa. Now the next possibility is that if all tail appears. All tail appears then person loses rupees so these are five amounts of money we can have after four tosses. Now we have to find the probability of having these amounts so in the first case e is the event of winning rupees four. So the person wins rupees four if he get all the head. Let's now see in how many cases he get all the heads so there is only one case in which he get a head and all the four tosses. So the number of outcomes favourable to e is one so the probability of winning rupees four is equal to the number of outcomes favourable to e which is one upon the total number of outcomes total number of outcomes are sixteen. Now in the second case we have to find the probability of winning one rupee fifty paisa so here e is the event of winning rupees one point five zero. Now the person win rupees one point five zero if he get three head and one tail. Let's now see in how many cases he get three head and one tail so either one two three and four so the number of outcomes favourable to e are four. So the probability of winning one rupee fifty paisa is the number of outcomes favourable to e which are four upon the total number of outcomes which are sixteen so the probability is one by four. Now in the third case we have to find the probability of losing rupee one so here e is the event of losing rupee one and in the case when he get two head and two tail he lose rupee one. And there are six such cases when he get two head and two tails so the probability of losing rupee one is equal to the number of outcomes favourable to e which are six upon the total number of outcomes which are sixteen so the probability is three by eight. Now in the next case we have to find the probability of losing three rupee fifty paisa so here e is the event of losing three rupee fifty paisa. The person loses three rupee fifty paisa when he get three tail and one head and in four such cases he get three tail and one head that means he lose three rupee fifty paisa. So the probability of losing three rupee fifty paisa is equal to the number of outcomes favourable to e that is the number of outcomes when the person get three tail and one head upon the total number of outcomes which are sixteen so the probability is one by four. Now in the next case we have to find the probability of losing rupee six so here e is the event of losing rupee six and the person loses rupee six if he get all the tails that he gets tail in all the four tosses and there is only one such possibility when he get all the four tails. So the probability of losing rupee six is equal to the number of outcomes favourable to e which is only one upon the total number of outcomes which are sixteen so the probability is one by sixteen. So there are five different amounts of money which a person can win or lose he can win rupee four which is a gain and he can win rupee one point five zero and he loses rupee one and he loses three rupee fifty paisa loses rupee six. So if you have obtained the probabilities of winning or losing these amounts this completes the question bye for now take care have a good day.