Mathematical Modeling Summer 2001

problems 0622

1. For a random walk with 6 steps, what distances from the center could occur and what is the probability of each possible distance?

2. Set up the transition matrix for a Sane-Demented situation with 7% of the sane becoming demented and 12% of the demented becoming sane. Use this transition matrix to find the populations after each of the first two transitions, assuming initial population of 500 sane and 500 demented.

3. If a population consists of sane, demented and borderline individuals, and if 10% of the sane and 20% of the demented becoming borderline during any transition period, while 15% of the borderline become sane and 10% of the borderline become demented, then if there are initially 500 sane, 500 demented and no borderline, how many of each of the three categories will there be after each of the first two transitions? Solve any way you wish, then if you haven't done so see if you can set up and solve using matrices.

4. A series of trees have populations 100, 300, 200, 100. There is a 10% transition to each adjacent tree during each transition, and in addition 5% of the bugs on the end tree get lost during each transition. Find the populations after each of the first three transitions, then if you haven't already done so see if you can use matrices to find the same results.

5. Repeat #4 assuming all the trees are in a circle.

6. There are initially 100 bugs on a tree in the middle of a long row of trees. During each transition the bugs each take a random walk of 2 steps. Give the population configuration of the nearby trees after each of the first two transitions.

7. Evaluate the function y = 2 x^2 – 3x + 5 for x = 1, 2, 3, 4, 5. Write your results in a sequence. Use successive differences to find the next two terms of the sequence. Then graph your sequence using Excel and fit a second-degree polynomial to your data, displaying the function. What function do you get?

8. Give the probability of each of the following:

• A billiard ball model with 4 balls in a rectangle will have all balls confined to the right half of the rectangle; the number of minutes during an hour of observation you would expect this configuration to be present.
• The same for 10, then 100, then 10,000 balls.
• The number of minutes in a billion years you would expect to observe every one of 1 million balls all in the right half of the rectangle.
• * The probability that, at any instant in the history of the universe, all the molecules in a typical room, anywhere in the universe, would all spend as much as 1 second all distributed on the right-hand side of the room, thereby causing severe respiratory distress to the occupants of the left-hand side of the room.
• ** The probability that any living entity the size of a mouse or larger, anywhere in the universe, has ever died as a result of such a phenomenon.

9. Suppose you have 8 trees in a circle, with initial population configuration given by the n=7 row of Pascal's Triangle. What do you think the population configuration will most likely be after each bug does a random walk of 3 steps?

10. Start with a number x between 0 and 1, noninclusive. Let m = 2. Calculate m x (1-x). Replace x with the result of your calculation. Repeat the calculation. Repeat for 10 steps. What is the resulting sequence of x values?

11. Repeat the preceding calculation for m = 3, then once more for m = 3.8.