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/