A Binomial Probability
Date: 04/10/99 at 18:29:37 From: Sharon Duncan Subject: Three Statistics Questions 1) I am trying to figure out how to do the "Monte Carlo Method." I have no idea how to begin. Can you help? 2) If I roll a balanced die 100 times, what is the probability I will get 14-21 ones inclusive? I know this is a binomial probabilty and I am stuck trying to set it up. We are working on normal distribution curves. The Ps is 1/6 and the Pf is 5/6. I know that n = 100 but how do I figure out the Mean and S.D? I know I have to make a .5 adjustment so the numbers I will be working with are 13.5 and 21.5. Thank you so much for your assistance.
Date: 04/12/99 at 14:14:47 From: Doctor Mitteldorf Subject: Re: Three Statistics Questions Dear Sharon, The Monte Carlo Method means something different for each question you ask. All it means is to simulate the process inside the computer, and see what happens. Usually this means thinking about the problem presented to you, using the computer's built-in random number generator to do something analogous to what's in the problem. For example, we'll take your question number 2. One way to answer that question is to do the binomial calculation, which I'll get to in a minute. You don't need a computer, and it's all algebra, and a lot of analysis and thought. The other way to do it is to use the computer to do it by brute force, and this requires much less analysis and thought, and no algebra. To do a Monte Carlo simulation of number 2, the core thing you'll ask the computer to do is Random(6). This function produces 6 different results with equal probability. Now put a loop around the process, running the function Random(6) 100 times. Count how many times the result came out to 1. This whole set of 100 simulated die-rolls is one trial. You can put ANOTHER loop around the program you've written so far, and run the entire 100-roll trial 10,000 times. Count how many of those trials come out 0,1,2,3... all the way up to 100. Of course, you'll find that 0 and 100 don't happen at all, and the numbers around 15,16,17 and 18 are quite common. After 10,000 trials you'll have a pretty accurate idea how many times the experiment generates results in the range 14-21. If you need more accuracy, just run the computer longer. At 1,000,000 trials you'll be 10 times more accurate than with only 10,000 trials. Moving on to the algebraic solution to number 2: There are two parts to the probability that get multiplied together to give your answer. One part is the product of individual events. If you're asking about 14, for example, 1/6 has to happen 14 times and 5/6 has to happen 86 times. So take (1/6)^14 * (5/6)^86. This is the first of the two terms you need to multiply together. The second term is the "number of ways" term: How many different selections of 14 objects are there out of 100 possible objects all together? That number is C(100,14), or 100!/(86!*14!). So the prescription is: The probability of getting 14 ones out of 100 rolls is (1/6)^14 * (5/6)^86 * 100!/(86!*14!). Add to this the probabilities for 15, 16, etc. all the way through 21, similarly computed, and you have an exact answer for the range 14...21. This is a lot of work on a calculator, but on a computer, it can be written up in a few minutes and will compute instantly. I hope this helps. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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