### Simulation Examples in EXCEL

```Simulation Examples in
EXCEL
Montana Going Green
2010
Example 1
• Flip of a fair coin
Algorithm
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Inputs: Number of trials (flips), N
Output: P(H), P(T)
Step 1: Set Counter H and T at 0.
Step 2: For I = 1 to N (number of flips) do
Step 3: Generate a random number, xi, between [0,1]
Step 4: If 0 < xi < .5, then H=H+1, else T=T+1
Step 5: Calculate P(H) = H/N and P(T)=T/N
Output P(H) and P(T)
Stop
Go to EXCEL
Example 2
• Area under a nonnegative curve such as
y=x3 from [0,2].
Algorithm
INPUT
The total number of random points, N. The nonnegative function, f(x),
the interval for x [a,b] and an interval for y [0,M] where M > max
f(x),a<x<b.
OUTPUT
The approximate area under the curve, f(x) over the interval [a,b]
Step 1. Specify the function, f(x) and set all counters at 0
Step 2.
For i from 1 to N do step 3-5
Step 3.
Calculate random coordinates in the rectangular region:
a<xi<b, 0<yi<M
Step 4.
Calculate f(xi)
Step 5.
Compare f(xi) and yi . If yi <f(xi) then increment counter by 1.
Otherwise, do not increment counter.
Step 6.
Stop
Estimate the area by A  M  (b  a ) 
counter
N
Let’s go to EXCEL
Example 3
Buffon Needle Experiment
• In mathematics, Buffon's needle problem is a question
first posed in the 18th century by Georges-Louis Leclerc,
Comte de Buffon:
– Suppose we have a floor made of parallel strips of wood, each
the same width, and we drop a needle onto the floor. What is the
probability that the needle will lie across a line between two
strips?
• Buffon's needle was the earliest problem in geometric
probability to be solved; it can be solved using integral
geometry. The solution, in the case where the needle is
not greater than the width of the strips, can be used to
design a Monte Carlo method for approximating the
number π.
Needle Experiment
Process
Buffon Needle Experiment
Let x be a uniform (0,1) that expresses the location of the end of the needle.
Let y be a uniform (o, pi) that gives the angle of the needle (0 is horizontal)
Let t = x+1*sin(y) be the horizontal projection of the needle
If t(i) > 1 Needle intersects line
Pi= 2 * (length of needle)/distance between parallel line * (number of tosses)/number crossing parallel line
If you allow the needle to be equal to the distance between parallel lines then Pi=2 * number of tosses/# crossing lines
Go to EXCEL
Fun Game
Monty Hall
• Here is the typical scenario. Monty has the
grand prize behind one of three doors. The
contestant picks a different door. Monty
choose the a different door (not yours and
not the grand prize). Do you stay with your
pick or switch to the other door?
• This question became a big discussion
• The problem was originally posed in a letter by Steve
Selvin to the American Statistician in 1975. A well-known
statement of the problem was published in Marilyn vos
1990:
• Suppose you're on a game show, and you're given the
choice of three doors: Behind one door is a car; behind
the others, goats. You pick a door, say No. 1, and the
host, who knows what's behind the doors, opens another
door, say No. 3, which has a goat. He then says to you,
"Do you want to pick door No. 2?" Is it to your advantage
• —Whitaker/vos Savant 1990
• Throughout the many years of Let's Make A Deal's popularity,
mathematicians have been fascinated with the possibilities
presented by the "Three Doors" ... and a mathematical urban
legend has developed surrounding "The Monty Hall Problem." The
CBS drama series NUMB3RS featured the Monty Hall Problem in
the final episode of its 2004-2005 season. The show's
mathematician offered his own, very definite solution to the problem
involving hidden cars and goats.
• The 2008 movie 21 opens with an M.I.T. math professor (played by
Kevin Spacey) using the Monty Hall Problem to explain
mathematical theories to his students. His lecture also includes the
popular "goats and cars behind three doors" example favored by
many versions of the Problem
Let’s go to EXCEL
Other problems
• Service with a smile: A small service department
has one server. Customers arrive according to
the distribution in Table 1. The server can
handle the customers according to the
distribution in Table 2. You want to use a
simulation to estimate the busy time of the
server, the average length of the queue, the
average waiting time of customers, and the
average number of customers present. Use a
simulation with one hundred events.
Tables for Problem
Table 1
Interarrival time
(minutes)
1
2
3
4
Probability
0.20
0.30
0.35
0.15
Table 2
Service time (minutes)
1
2
3
Probability
0.35
0.40
0.25
```