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### A Binomial Probability

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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.
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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
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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Associated Topics:
High School Probability

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