3. Find the expected value and variance for the evening's gain, the sum of the gains or losses for the 200 bets of $5 each. The technique of simulating a process that contains random elements and repeating the process over and over to see how it behaves is called a Monte Carlo procedure. It is widely used in business and other fields to investigate the properties of an operation that is subject to random effects, such as weather, human behaviour, and so on. For example, you could model the behaviour of a manufacturing company's inventory by creating, on paper, daily arrivals and departures of manufactured products from the company's warehouse. Each day a random number of items produced by the company would be received into inventory. Similarly, each day a random number of orders of varying random sizes would be shipped. Based on the input and output of items, you could calculate the inventory that is, the number of items on hand at the end of each day. The values of the random variables, the number of items produced, the number of orders, and the number of items per order needed for each day's simulation would be obtained from theoretical distributions of observations that closely model the corresponding distributions of the variables that have been observed over time in the manufacturing operation. By repeating the simulation of the supply, the shipping. and the calculation of daily inventory for a large number of days (a sampling of what might really happen), you can observe the behaviour of the plant's daily inventory. The Monte Carlo procedure is particularly valuable because it enables the manufac- turer to see how the daily inventory would behave when certain changes are made in the supply pattern or in some other aspect of the operation that could be controlled. In an article entitled "The Road to Monte Carlo," Daniel Seligman comments on the Monte Carlo method, noting that, although the technique is widely used in busi- ness schools to study capital budgeting, inventory planning, and cash flow manage- ment, no one seems to have used the procedure to study how well we might do if we were to gamble at Monte Carlo.