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Math 170 Lecture Notes: Ideas in Math - Probability & Dynamical Systems, Study notes of Mathematics

These lecture notes provide an introduction to probability theory and dynamical systems. The concepts of union of probabilities, non-disjoint events, and the infinite monkey theorem. It also introduces the idea of equilibrium in dynamical systems and discusses stable and unstable equilibria.

Typology: Study notes

2009/2010

Uploaded on 03/28/2010

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koofers-user-lpe 🇺🇸

10 documents

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Lecture Notes for Math 170: Ideas in

Mathematics (Spring 2007)

by Nathanael Leedom Ackerman January 17, 2007

1

1 TALK SLOWLY AND WRITE NEATLY AND BIG!! 2

1 TALK SLOWLY AND WRITE NEATLY AND BIG!!

2 Misc Collect Homework

Mention that the lecture notes are just what I do for myself and they are not meant to be taken as correct. They are mainly there just to give you a sense of what was covered in the class.

3 Review From Last Time Go through the proof using marbles again on conditional probabilities (i.e. draw a rectangle with every possible combination of marbles from two bags and observe that the top left corner is the marbles we want).

Two possibilities

4 Union of Probabilities

These marbles also allow us to see another of the most important theorems of probability.

4 UNION OF PROBABILITIES 4

With these two theorems we can calculate the probability of almost anything.

Non-disjoint events To see this lets try

and calculate the probability of two events happening if it is possible for them both to happen.

Suppose we have a bag of 100 marbles each labeled with a number and we want to calculate the chance of drawing a multiple of 2 or a multiple of 3.

We know there are 3 possible disjoint outcomes which satisfy the condition we are looking for. So we want to find

(A) The number of marble which are a multiple of 2 and not 3

4 UNION OF PROBABILITIES 5

(B) The number of marble which are a multiple of 3 and not 2

(C) The number of marble which are a multiple of both 2 and 3

Now what is interesting is that

  • Number of multiples of 2 = A + C
  • Number of multiples of 3 = B + C

So A + B + C which is what we are looking for is Number of multiples of 2 + Number of multiples of 3 - C.

So we have A + B + C = 50 + 33 − 16 = 67.

In general we have

Theorem 4.0.2. If A and B are two events, then the

5 DYNAMICAL SYSTEMS 7

Theorem 5.0.4. Let E be an outcome of an event O such that the probability of E < 1. Then for all real numbers ≤ there is an n such that if you repeat event O n times the odds of E happening every single time less than ≤.

Specifically if we repeat event O n times then the odds of E happening every time is P rob(E)n.

So, if P rob(E) < 1 then limn→∞ P rob(E)n^ = 0.

Now one of the things which is interesting about this that if P rob(E) = 1 then we know two things.

  • P rob(E)n^ = 1
  • limn→∞ P rob(E)n^ = 0

Discrete Dynamical Systems We can

view this repeated event O as what is called a discrete dy-

5 DYNAMICAL SYSTEMS 8

namical systems. The idea is that we approximate time by a series of steps and at each step the state of our sys- tem changes.

So let Pn be the probability that event E has occurred every time during the first n time steps. Then Pn+1 = Pn ∗ P rob(E).

Definition 5.0.5. Let Pn+1 = f (Pn). We then say a state a is an equilibrium if f (a) = a.

So if a systems we are modeling is in an equilibrium state it doesn’t change.

Lets give another model which will make it a little more clear what is going on.

6 TYPES OF EQUILIBRIUM 10

we start with almost 0 dollars. Will this bank grow your money or not?

Well it isn’t hard to see that Pn = P 02 n. (Write out why). So we find that if we start with anything less than 1 dollar our money will eventually go to 0.

Definition 6.0.6. We say a is a stable equilibrium or an attractor of Pn = f (Pn− 1 ) if f (a) = a and for all sufficiently small ≤ P 0 = a + ≤ implies limn→∞ Pn = a

The idea is that if we start close to a stable equilibrium we will eventually end up there.

Pn = 1: We have already seen that if we start at 1 − ≤ (where ≤

6 TYPES OF EQUILIBRIUM 11

is positive) then our limit will be 0. What if we start at 1 + ≤?

Well it isn’t hard to see that in this case we have limn→∞ Pn = ∞ 6 = 1. This is an example of an unstable equilibrium.

Definition 6.0.7. We say a is an unstable equilibrium of Pn = f (Pn− 1 ) if f (a) = a and for all sufficiently small ≤ P 0 = a + ≤ implies limn→∞ Pn 6 = a Ask if there are any other types of equilibrium.

Give example of a magenent on the side of a cliff where the system is a metal ball. So on once side it is stable and on the other sid the ball falls off the cliff) it is unstable.

If still time

Give example of a pendulum. Give Quiz