Please submit your excel sheet, so I can look at it, but put your answers in the text of your e-mail, so that if you get all the right answers I don't have to look at your work. Thanks!
1. (2 pts) Assume you are rolling three dice at the same time. What is the probability that none of them will come up six?
2. (5 pts) Now, set that problem up in Crystal Ball. The easiest way to do it is to use Excel's "if" statement in your forecast cell, and have it output a 1 if any cell has a 6 in it. There are two ways to do this: one would be to have three if statements in different cells, and then have your forecast cell be the sum of these three cells. Then, if your forecast cell is a value greater than 0, you know at least one of your results was a six. Even easier is to use nested "if" statements, although some people get confused by these. For example, let's say that you have some data in cell "A3" that you want to print only if the values in "B3" and "C3" were both greater than zero. If not, you want Excel to print out a "0". The way you would do this would be to put in cell D3 something like this:
=if(B3>0,if(C3>0,A3,0),0)
Notice that the second "if" statement is in the "true" space for the first "if" statement. The way the computer interprets this line is that it asks itself "Is B3 greater than zero?" If the answer is "No", then it fills the cell with the value 0 (The last 0 there, right before the final paren). If the answer is "Yes", then it goes to the "true" space, which is the middle value. But when it goes to figure out what to print out, it finds that it needs to execute another "if" statement, namely: "If C3>0, print A3, otherwise print 0". In your problem, you will probably want to tell Excel to ask "If A3 is 6, print 1, otherwise, if B3 is 6, print one, otherwise, if C3 is 6, print one, otherwise print zero". See if you can convert that into an if-then statement. Then, run the simulation. How does your result compare to the calculated result?
3. (3 pts) Now, let's see how long it takes for Crystal Ball to get a reasonable answer. One thing we need to know when using Monte Carlo is when we are close enough. If we don't run enough simulations, then we'll come up with an incorrect result. If we run too many, then we'll spend all day running a few simulations, and our bosses will think we're slacking off, or we won't get the results in time and someone else will publish before we do, and we won't get tenure. So, getting the number of runs right is very important, and it changes for different types of runs. Usually we need to do somewhere between 1000 to 10,000 runs. There are a couple of ways to determine how many is the "right" number of runs, but one way that works if we are going to do a lot of runs is just to make a graph for a typical run and use it to set your preferences for all future runs. That is what I want you to do here.
First, you will need to make sure that all of your runs go through the same set of "random" numbers, so that you can see how long it takes for early anomalies to die out. To do this, you need to "seed" your generator. Go to the "Run", "Run preferences", and select the "Sampling" tab. Check the box that asks about the same sequence, and give it a seed of 1 (you can use any integer above 0, but we'll go with 1). Now, do the run for 10, 100, 500, 1000, 2500, 5000, 7500, and 10,000 trials, and mark down the probability that you get for each run. Make a chart, where you graph the probability versus the number of trials. Looking at your graph, how many trials do you think you need to run to get the "right" answer? Put another way, if you had to analyze a few hundred questions regarding probabilities like this one, how many runs would you use for each analysis?
4. (5 pts) Pretend that you are working on a project where you need to know about how far butterflies will fly, and in which direction. Download the following two text files:
Butterfly_distance.txt (data in cm)
Butterfly_direction.txt (data in degrees)
This is real data from a project some of us are working on. Now, what type of distribution best describes the distance that butterflies fly? What type of distribution best describes the direction at which they fly away?
5. (5 pts) What is the probability that a butterfly which starts at a given point will be found 400 to 600 cm to the northwest of its starting point, after one movment?