Harry Marshall
Group B
02/29/2007
Area of Diamond Assignment
N(r) = 2R+1 + 2r^2 and B(r) = 4r
Feb 28 Assignment: describe Pick’s theorem and check if it is consistent with the N(r)
and B(r) results
Pick's theorem
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with
integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem
provides a simple formula for calculating the area A of this polygon in terms of the
number i of interior points located in the polygon and the number b of boundary points
placed on the polygon's perimeter:
A = i + ½b − 1.
Last week, we concluded on the Area of Diamond Assignment, that:
N(r) = 2R+1 + 2r^2 and B(r) = 4r
So, substituting b with 4r and substituting i with 2r+1+2r^2 -4r (we take away 4r so we
have only interior points) which simplifies to 2r^2-2r +1, we arrive at Pick’s Theorem
specifically for our diamond. That is:
A= 2r^2-2r +1 + ½ (4r) -1 or
A= 2r^2-2r+2r or
A= 2r^2
In our diamond last week, we deduced that area of “diamond” = 4 right isosceles triangles
A = 4 × ½r × r
Or
A = 2r^2
So Pick’s Theorem is consistent with our N(r) , B(r) results for the diamond 
1
Harry Marshall
Group B
02/29/2007
Feb 14: 1-D discrete Random Walks in Excel
generate and graph a single 10-step walk
add 9 more 10-step walks to the graph
optional: count the number of walks that finish at each end position
=COUNTIF(range ,value)
From last week,
Feb 28 extension assignment:
Based on this single experiment’s count, what is the probability of reaching each
endpoint? 0%
What is the theoretical probability? 10C10/2^10
.00098
Plot bar graphs of both of the above.
2
Harry Marshall
Group B
02/29/2007
Coin Flipping experiments in Excel
If N=100 fair coins are flipped (or 1 fair coin is flipped 100 times) how many are heads?
50
Feb 28 assignment: Coin Flipping in Excel
Complete the Coin Flipping spreadsheet shown in class and document the construction
and discuss what it demonstrates in your narrative.
First, I typed the following with static text:
3
Harry Marshall
Group B
02/29/2007
Then, I renamed cell B2 offset. . .
and plugged the formula, =INT(RAND()+offset) into cell C4.
I, then, dragged the formula across and down to fill up the yellow grid area . . .
I entered a Countif formula into cell O4 to count the number of cells in my grid (which I
selected by dragging over) = 1.
4
Harry Marshall
Group B
02/29/2007
I entered the formula =100-O4 into cell O5 but could have just as easily made another
Countif statement to count the number of 0’s in my grid.
I entered the following text:
and resized my graph so the text would fit on my screen. I entered the formula =100-C16
into cell C17 and dragged the formula across so when I performed my experiments, I
would only have to enter the Heads values.
I entered the first value of heads, 48, and the spreadsheet changed. I continued entering
the next 19 results by pressing tab to move to the next cell to the right. The results are
below:
I used Excel to find the average by summing
the results of the heads experiment and
dividing by 20 in cell C19 and typing =100C19 into cell C20. The results are pictured:
I
5
Harry Marshall
Group B
02/29/2007
used Microsoft Help to derive the correct formula for standard deviation
And used it to calculate the standard deviation of the entire set (population) of results.
The standard deviation is a statistic that tells you how
tightly all the various examples are clustered around
the mean in a set of data. When the examples are
pretty tightly bunched together and the bell-shaped
curve is steep, the standard deviation is small. When
the examples are spread apart and the bell curve is
relatively flat, that tells you you have a relatively large
standard deviation.
http://www.robertniles.com/stats/stdev.shtml
6
Harry Marshall
Group B
02/29/2007
When the offset value was changed to .75, the results clustered around 75 heads and 25
tails but the standard deviation did not change much. See below:
What is now the probability of a single flip giving heads (1)? 75%
With this probability for a single flip, what is the probability that two flips will give the
same result (both heads or both tails)? .75 x .75 + .25 x .25 = .5625 + .0625 = .625 or
62.5%.
What is the probability of getting NH heads when a fair coin is flipped N times?
=COMBIN(N, NH)/2^N
7
Harry Marshall
Group B
02/29/2007
Discuss how this function varies as N increases:
Either generate graphs of this probability for N = 10, 20, 40, 100 or use the graph on the
next slide as the basis for the discussion.
What happens to the average? As N increases, the average number of heads flips gets
closer to ½ N.
What happens to the width of most probable outcomes about the average (standard
deviation)? It expands to include more of the actual results clustered around the
theoretical outcomes. As N approaches infinity, the distribution will approach that of a
normal bell curve with 99% of results falling within 1 standard deviation from NH, 95%
falling within 2 standard deviations, and 86% falling within 3 and the differences
between the actual and theoretical probabilities become negligible.
8