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Student Answers to Questions

1. "The probability-r in the number-n row of Pascal's Triangle tells us how many of these outcomes will have r heads."

I am not absolutely certain what this is saying.

It's saying that the number on which ever row of Pascal's Triangle represents the probability of anything, for example the coin flips and the random walk.

For a random walk of 3 steps, you can land on 1 and 3. the third row of pascal's triangle is 1 3 3 1. there is 2 ways of landing 1 steps away, and there is 6 ways of landing 6 steps away.

What is meant by row number i for transition matrices? And what is a column vector?

'i' represents any row in a matrix, the column vector is what the matrix is multiplied by. It is part of the transition matrix. The answer is also a column vector.

40. What is "cells" or lattice point in diffusion?

"Cells" are referring to cells in Excel.

3.Random Walks

What does "r" represent?

* "r" simply represents the number of steps taken to the right

42. How to use "e" on the calculator:

On most calculators, "e" will be shown above a button

(not any button in particular - it varies from calculator to calculator).

Just scan your calculator for the symbol e^x. When you find it, hit the "2nd" button and then the button that has "e^x" above it. This will show you e^( .

Now apply what you know:

For example,

Your problem tells you that e^(6).

So just put 6 after e^( and then close the parenthesis.

Like this: e^(6)

Then hit ENTER.

* Please excuse the over-use of the word "button".

Question 18. Remember the Christmas Tree problem? That's Cellular Automata.

I do not know how to use e on the calculator.

If you are using a scientific calculator, first you need to enter the exponent you are raising e to. Then, press e. For example, if you want to calculate e^(3*5), you would multiply 3*5 and after you get the answer, press e.

If you are using a graphing calculator, press e and you will see e^( on the screen. Simply enter the number you want to raise e to and close the parenthesis, then press enter or execute. Example: e^(3*5)

Question 34. Diffusion is where you have something moving along something else, Such has bugs moving between trees.

Q) Trapezoidal Graphs

I know what altitude is. However, I do not understand how to find it for the trapezoid.

A) Altitude is the difference from the given point to the x-axis. For example if a given point was at coordinates (3,5) then you were to draw a line from the point, straight down to the x-axis, then the line would be 5 units long, so the altitude of the line would be 5 units. Now to find the are of the trapezoid, you must find the average altitude (add the two altitudes up and divide by two),and multiply this by the width of the trapezoid (the difference one the x-axis, between the two points).

Trapezoidal Graphs

What does "r" represent?

I believe "r" represents right, as in the direction.

Question 72. Cellular Automata is where something switches itself according to the things or environment around it, as in the Christmas Tree problem.

16. I understand how to set up the Trapezoidal Graphs and how to graph then, but I still don't understand how to find the slope and the average area. Could you please explain how to find these things?

After you graph the points and draw the lines to make the graph look like a trapezoid , you want to find the slope and the area of the space in order to solve the problem. In order to find the slope of the trapezoid, you use rise over run. The difference in the altitude between the two points, over the distance between the two sides.

Say you have a graph in which one side is 11 units, and the other side is 16 units, and the distance between them is .5. The rise is 5 (the distance in the altitudes), and the run is .5 (the distance between the two sides. Therefore your slope is 5/.5, which equals 10.

To find the area, you use the average altitude * the width. Which is ((11+16)/2)*.5, which equals 6.75.

Question # 31

A random walk is when someone or thing is walking around without knowing where they are going. For example if I didn't know where a store I was looking for was I might take a couple steps to the right and look down that way for a while. Then I might decide that I didn't see the store and take a couple steps to the left and continue looking. That would be a random walk because I would be guessing at which side it could be on.

What is diffusion?

Diffusion is the spreading out of molecules, bugs, etc. from an area of high concentration to an area of low concentration.

31. What is a random walk?

A random walk is simply a step in any possible direction.

Do you recall the coin flip experiment that we did in the hall?

Well, those random steps that you took right and left are basically the same steps that you are taking on the x and y axis.

Question # 95

S' is to find a pattern between a sequence of numbers. The S' is the first row of numbers you have and then to find the pattern of the sequence of numbers you use the second row S'' that is really all that means. You can then use this set of data to find the answer to the question you need to work. The S' helped you to find the pattern of the sequence of numbers so you could maybe graph these or use the pattern to find the next number in a sequence.

Question 76. Okay, the list of two flips HH, HL, LH, and LL, right? Now you can add an L or H to every one of those to get the list of three flips which is HHH, LHH, LHL, HHL, HLH, LLH, HLL, and LLL.

Question # 42

To use "e" on the calculator first type the exponent of "e" into the calculator. Then hit the second key (usually a yellow key on Calc.) and hit the button that has "e" above it and then it the "*" key and then the number that you want to multiply "e" by. That is how to use the "e" key on the calculator.

34. What is diffusion?

Diffusion is the transfer or movement of something from one place to another.

What meant by "Row number I"?

I believe he means row 1.

1. Pascal's Triangle and Probability

2. Don't understand- how you can use the list for 2 flips to make a list of 3 flips.

Don't know how to count the number of ways to get heads.

For two flips, the probability would be 1/4, 2/4, 1/4,

For zero head you get 1/4

For 1 head you get 2/4

For 2 heads you get 1/4

For three flips use fourth of pascal's triangle

A random walk is what it sounds like, you just take a step left or right depending on if you get heads or tails by flipping a coin.

What is a random walk?

A random walk is a series of random movements in an x- direction which can be considered a number of movements along a straight line which only the probabilities can be predicted using pascal's triangle.

To find the altitude of a trapezoid, you average the altitude of both sides.

1. Trapezoidal Graphs

2. I know what altitude is. However, I do not understand how to find it for the trapezoid.

The 2 lines drawn from the points down to the x axis form the 2 bases for the trapezoid. The altitude is the distance on the x axis that separates the 2 bases.

To find the slope of a trapezoid graph you have to first take the rise which is the amount of distance that the line goes up or down between data points and take that number and divide it by the run which is the width of the trapezoid.

To find the average area of the trapezoid you take the height at the first data point and then take the height at the second data point on the trapezoid add them together and divide by two, this will give you your average altitude. Then take the average altitude and multiply by the trapezoids width and this will give you the average area of the trapezoid.

1. Exponential Population Growth

2. What does "k" represent? Also, I am having a hard time reading this. It's a little confusing

k represents a number that can have any value required by the situation. e^(.01 t) and e^(3 t) are both exponential functions, but the graph of the first is much 'flatter' than the graph of the second.

Question 11. yes