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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
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by Nathanael Leedom Ackerman January 17, 2007
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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).
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.
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
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.
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.