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Article: The relationship between the binomial expansion series, trees, probability, and counting 5102005
The relationship between the binomial expansion series, trees, probability, and counting
If you toss a coin into the air, it can land either heads or tails. If you write H for Heads
and T for tails, you can draw the number of possible outcomes like this:
This diagram is called a "TREE". Each branch of the tree represents a different possible
outcome from tossing a coin twice into the air. There are 2 possible outcomes from tossing
a single coin into the air, and so there are 2 branches on this "Tree".
Below is a table summarizing the relationship between 1 coin toss, a tree, and the
binomial expansion series from algebra:
1 coin toss


From Algebra, remember that (H + T)^{1} = H + T

If you toss a coin into the air twice in a row, how many different possible ways can
the coin land?
From Algebra, remember that (H + T)^{2} = (H + T) * (H + T) = (H * H) + (H * T) + (T * H) + (T * T)
and
(H * H) + (H * T) + (T * H) + (T * T) = H^{2} + 2HT + T^{2}
Also notice that in algebra, HT "MEANS" H times T...So if you remove the * symbol you get:
(H * H) + (H * T) + (T * H) + (T * T) = HH + HT + TH + TT
A Pattern: Notice in the diagram below that
HH corresponds to the right most branch of the tree,
HT to the 2nd from the right branch of the tree,
TH to the 2nd from the left branch of the tree,
and TT to the left most branch of the tree
2 coin tosses


(H + T)^{2} = HH + HT + TH + TT

Each TERM in the binomial expansion represents a different branch on the tree.
Furthermore, you can see that (H + T)^{2} physically MEANS "tossing
a coin into the air twice in a row, or tossing 2 coins into the air simultaneously".
After 3 coin tosses
(H + T)^{3} = HHH + HHT + HTH + HTT + THH + THT + TTH TTT
3 coin tosses


After 4 coin tosses
(H + T)^{4} = HHHH + HHHT + HHTH + HHTT + HTHH + HTHT + HTTH + HTTT + THHH + THTH + THTT + TTHH + TTHT + TTTH + TTTT
4 coin tosses


Trees and Information
Say you are travelling on vacation and you get lost. What does it mean to get lost?
It means you are uncertain about the direction you are to go. Do you turn right
or left at the stop sign? You could simply guess which way to go, or you could stop and
ask for directions. If you guess, you have a 50% chance of being correct, or a probability
of 1/2.
If you stop and ask someone if you should turn right or left, you will have aquired information.
Similarly, if you guess that you should turn left and you discover that you have guessed
incorrectly, you will have also aquired information (You now know that the correct direction
to take at the traffic light is a right hand turn).
So, in some way, information is related to uncertainty(or probability), and aquiring information
is to eliminate the uncertainty that you had(or ruling out possibilities).
This relationship was discovered by a man named Claude Shannon, and it is called
INFORMATION THEORY. Click here to learn more about what probability and uncertainty have to do with the
bits and bytes that are found inside of a computer!
The relationship between the binomial series, area, and volume
If you have a square with sides of length A,
The area of this square is A * A = A^{2}.
If you increase the length of each side of the square by a distance of B,
so that each side of the square is now A + B, its new area is
(A + B) * (A + B) = (A + B)^{2}
Adding the area of each of the above squares gives you the total area of the square
whose sides are length A + B. This area is AA + AB + BA + BB
AA + AB + BA + BB = A^{2} + 2AB + B^{2}
If you have a cube whose sides are all A, its
volume is A * A * A = A^{3}
If you increase the length of each side of the cube by a distance of B,
so that each side of the cube is now A + B, its new volume is
(A + B) * (A + B) * (A + B) = (A + B)^{3}
Can you see that the cube above is made up of 8 smaller cubes, and that the sum
of the volume of these 8 smaller cubes EQUALS (A + B) * (A + B) * (A + B) = (A + B)^{3}
The binomial series can be thought of as both the tossing of a coin AND the surface
area of a square or volume of a cube!!!
4 dimensional objects and hypercubes
Question:
If (A + B) is the length of a line
and (A + B)^{2} is the area of a square
and (A + B)^{3} is the formula for the volume of a cube,
then does (A + B)^{4} have any physical, geometric interpretation? What about
(A + B)^{5}, (A + B)^{6} or higher?
It turns out that it does!
Click here to learn about the 4th dimension and hypercubes

