 Hello and welcome to the session. In this session we will discuss a question which says that an organization refers a bike worth 5000 dollars to raise money and wants to decide that which of the following situations would earn more gain on the expected value of the proposed case A is 50 tickets are sold at 200 dollars each and case 3 is 90 tickets are sold at 150 dollars each. Now let us start with the solution of the given question. Now in this question we are given that an organization refers a bike worth 5000 dollars to raise money also we are given two situations that is these two situations and we want to determine that which situation would earn more gain based on the expected value of the proposed gain. Now here in both the cases we will determine the expected value of winning of the players. Now case A is given as 50 tickets are sold at 200 dollars each. Now let us find random variable values for gain that is we have to find the X values. Now the organization refers a bike worth 5000 dollars and cost of one ticket is 200 dollars if the player wins he will get a bike worth 5000 dollars. This means his gain is 5000 dollars and if he loses then his gain will be minus 200 dollars and that is loss of 200 dollars which is the value of the ticket. Now let us make a table for random variable values X and its probability P of X. So here we have made this table and we will write the corresponding values. Here as we have already discussed that if the player wins he will get a bike worth 5000 dollars. So here X value is if the player wins probability will be 1 upon 50 because for case A the total number of tickets is equal to 50 and as here bought one ticket so for winning probability is equal to 1 upon 50. So here when the player wins the probability is 1 upon 50 and if the player loses then his gain will be minus 200 dollars and its probability will be 49 upon 50 because here for winning as the probability is 1 upon 50. So for losing the probability will be 1 minus 1 upon 50 that is equal to 49 upon 50. Now from this table we can find the expected value of winning. Now we know that expected value of winning is given by summation of X into P of X the game and from this table we know the values of X and the corresponding values of P of X. So the expected value is equal to 5000 dollars into its probability that is 1 upon 50 plus of minus 200 dollars into 49 upon 50 and this is equal to now on solving this will be 100 dollars minus 196 dollars and this is equal to minus 96 dollars. So for first case the expected value of winning is equal to minus 96 dollars. Now let us take the second case and it is 90 tickets are sold at 150 dollars each. Now in case B also will win a table for the X values its probability. So here we have made a table for finding random variable values or X values for gain and for finding the values of P of X also. Now in case B 90 tickets are sold at 150 dollars each it means a player bought a ticket for 150 dollars. So if he wins he will get a bike worth 5000 dollars so his gain is 5000 dollars and if he loses his gain will be minus 150 dollars that is the cost of a ticket that is loss of 150 dollars. Now if he wins the X value is 5000 dollars and its probability will be 1 upon 90 and if he loses then his gain will be minus 150 dollars and its probability will be 89 upon 90 which is equal to 90 and as a player bought one ticket so for winning probability will be 1 upon 90 and for losing probability will be 89 upon 90. Now in this case also using the values given in this table we can find the expected value of winning that is equal to summation of X into P of X the whole. So here expected value is equal to 5000 dollars into 1 upon 90 plus or minus 150 dollars into 89 upon minus 148.33 dollars and this is equal to minus 92.83 dollars so for case A expected value is minus 96 dollars and for case B it is minus 92.83 dollars that means in case A where he is expected to gain minus 96 dollars and if he wins for case B then he is expected to gain minus 92.83 dollars. The expected loss for the player in case A is greater than the expected loss for the player in case B. So the organization will raise more money because in case A the greater expected loss for the player assures that the organization has a higher expected gain. So this is the solution of the given question that's all for this session hope you all have enjoyed the session.