# Category Archives: Algebra of events

## Outcome space

Let $$\Omega$$ be an outcome space with 16 outcomes. $$A$$ and $$B$$ are events inside of $$\Omega$$. Event $$A$$ has 10 outcomes and event $$B$$ has 10 outcomes.

1. Determine all the possible values of $$\# (A\cap B).$$
2. Determine all the possible values of $$\# (A\cup B).$$
3. Determine all the possible values of $$\#(A^c\cup B^c).$$
4. Determine all the possible values of $$\# (A^c\cap B^c).$$

## Algebras and Conditioning

Consider a deck with three cards number 1, 2,  and 3. Furthermore, assume that 1 and 2 cards are colored red and the 3 card is colored black. Two of the cards are drawn with out replacement. Let $$D_1$$  be the first card drawn and $$D_2$$ be the second card drawn. Let $$T$$ be the sum of the two cards drawn and let $$N$$ be the number of red cards drawn.

1. Write down the algebra of all possible event on this probability space.
2. What is the algebra of events generated by $$T$$, which we denote will $$\mathcal{A}(T)$$ ?
3. What is the algebra of events generated by $$N$$ , which we denote will $$\mathcal{A}(N)$$ ?
4. Is $$T$$ adapted to  $$\mathcal{A}(N)$$ ? explain in terms of the above algebras.
5. Is $$N$$ adapted to  $$\mathcal{A}(T)$$ ? explain in terms of the above algebras.
6. What is $\mathbf{E} [ N \,|\, \mathcal{A}(T)] ?$
7. What is $\mathbf{E} [ T \,|\, \mathcal{A}(N)] ?$

## Algebras and Conditioning

Consider two draws from a box with replacement contain 1 red ball and 3 blue balls. Let $$X$$ be number of red balls. Let $$Y$$ be 1 if the two balls are the same color and 0 otherwise. Let $$Z_i$$ be the random variable which returns 1 if the $$i$$-th ball is red.

1. What is the sample space.
2. Write down the algebra of all events on this sample space.
3. What is the algebra of events generated by $$X$$ ?
4. What is the algebra of events generated by $$Y$$ ?
5. What is the algebra of events generated by $$Z_1$$ ?
6. What is the algebra of events generated by $$Z_2$$ ?
7. Which random variables are determined by an another of  the random variables. Why ? How is this reflected in the algebras ?
8. (*) What pair of random variables are independent ? How is this reflected in the algebras ?

## Counting and geometry

1) How many ways can one order $$n$$ people on a line ?

2) How many ways can one order $$n$$ people on a circle or round table (note that rotations are considered equivalent )?

3)  How many ways can one order $$n$$ people on a circle or round table with invariance with respect to rotations and direction (clockwise versus counter-clockwise) ?

## Algebra of events

Consider the sample space created by two coin flips. We will denote the outcome of first a head and then a tail by $$(H,T)$$. The even that there is exactly one head will be written $$\{(H,T), (T,H)\}$$.  Let $$\mathcal{A}$$ denote the set of all events on this sample space.

1. Write down all of the events in $$\mathcal{A}$$. How many are there ?
2. Pick a few elements of $$\mathcal{A}$$ and show that the rules of being an algebra are in fact satisfied.  Namely, if $$A,B \in \mathcal{A}$$  then $$A \cup B \in \mathcal{A}$$ and $$A^c \in \mathcal{A}$$.  (Here $$A^c$$ is the compliment of $$A$$ which is simply every event which is not is $$A$$.)
3. Why did I not ask you to verify by hand this property for all of the sets in $$\mathcal{A}$$ ? Hint: can you tell me a lower bound on the number of such conditions  you would have to check ?
4. (*) Using the definition of an algebra, show that the empty set $$\emptyset$$ is in $$\mathcal{A}$$.
5. (*) Using the definition of an algebra, show that  if $$A, B \in\mathcal{A}$$ then so must be $$A \cap B$$.
6. (*) Argue that if $$A_1, A_2,\dots,A_k \in\mathcal{A}$$ then so are
$A_1\cup A_2\cup\cdots\cup A_k=\bigcup_{j=1}^k A_j$
$A_1\cap A_2\cap\cdots\cap A_k=\bigcap_{j=1}^k A_j$