Finite Final

x={red, blue, yellow}
yellow E x
"yellow is an element of x"
one element in a set

x={red, blue, yellow}
{red, blue} C x
"red and blue are a subset of x"
a set contained within a set

A U B
merging two sets together
in A or in B

elements in A AND in B
must be in both to be correct

A'
set of elements in the universal set that aren't in A

A' n B'= (A U B)'
A' U B'= (A n B)'

A={Chicago, Indy, Milwaukee} and B={Detriot, St.Louis}
A x B={(C,D), (C,S), (I,D), (I,S), (M,D), (M,S)}
B x A= {(D,C), (D,I), (D,M), (S,C), (S,I), (S,M)}
they are not equal

when two sets have no elements in common

every set in the collection is disjoint from every other set in the collection

collection of pairwise, disjoint, non-empty sets who union is the universal set

2^n

n(A x B)= n(A) x n(B)

n(A U B)=n(A) + n(B) - n(A n B)

n(A U B U C)=n(A) + n(B) + n(C) - n(A n B) - n(A n C) - n(B n C) + n(A n B n C)

activity that produces two or more possible outcomes
ex:
flipping a coin, rolling a die, playing a chess match

what the experiment produces
ex:
2 outcomes in flipping a coin
6 outcomes in rolling a die

possible outcomes of an experiment put into a set
ex:
(heads, tails) in flipping a coin
(1,2,3,4,5,6) in rolling a die

when a given stage does depend on choices made earlier
ex:
A license plate number consists of 4 letters of the alphabet following by two of the digits 0-9. The letters I and O are not allowed. How big is the sample space?
L L L L D D
24 24 24 24 10 10 = 24^4 x 10^2

subset of the sample space of an experiment
ex:
"rolling a total of 5" when two die are rolled
{(4,1), (3,2), (2,3), (1,4)} 4/36

**weights must add up to 1**
ex:
sample space contains 5 outcomes (O1, O2, O3, O4, O5). The weights are w1= .3, w2= .1, w3= 2w5, w4= w1+2w5. What are the values of all the weights?
w1= .3
w2= .1
w3= .12
w4= .42
w5= .06

ex:
What is the probability of the event {O2, O5}?
w2+w5= .16

Pr(E)= n(E)/n(S)

doing something with repeated stages

P(n,r)= n!/(n-r)!
- count the number of ways of both selecting and assigning items to specific roles

C(n,r)= n!/(n-r)! r!
- count the number of ways of selecting some number of items
ex:
How many Poker hands contain at least one Ace?
C(4,1)xC(48,4) + C(4,2)xC(48,3) + C(4,3)xC(48,2) + C(4,4)xC(48,1)

1

1

n

permutations: counting all the possible arrangements of the chosen elements, selecting and arranging

combinations: different arrangements of the selected elements are not considered distinct, selecting but NOT arranging

both can't happen at the same time

two events are independent when the probability that both events occur is the product of their probabilities
ex:
Pr[E]=.4
Pr[F]=.5
Pr[E n F]=.2

the conditional probability of E given F

Pr[EIF]= Pr[E n F]/Pr[F]

when there are two possible outcomes

C(n,r) x p^r x q^n-r

the numerical quantity associated with the outcomes of an experiment

random variable1 x probability1 + random variable2 x probability1 + .....

*when you have to write out the chart with event and probability of it

E(x)= np

ex. a weighted coin with probability Pr[H]=.4 is flipped 10 times. What is the expected value of the number of heads?
n x p
10 x .4= 4

m= riseun

y= mx+b
*use when you have y-intercept and slope

y-y1= y2-y1/x2-x1 (x-x1)
*use when you have two points

y-y1= m(x-x1)
*use when you have a point and slope

Ax+By=C

horizontal line with a slope of 0

vertical line with an undefined slope

1. reorder the equations
2. multiply an equation by an nonzero number
3. add or subtract any two equations

(1/ad-bc) d -b
-c a

superaugment matrix A and I
simplified matrix on the right is the inverse matrix

coal: X(production of coal)=Acc(internal demand for coal)+Asc(internal demand for steel)+Dc(coal for external demand)

steel: X(production of steel)=Ass(internal demand for steel)+Acs(internal demand for coal)+Ds(steel for external demand)

matrix X= production of coal
production of steel

matrix D= external demand for coal
external demand for steel

matrix A= amount C needed for C amount C needed for S
amount S needed for S amount S needed for C

X= (I-A)^-1 x D

(I-A)
1 0 - .3 .4 = .7 -.4
0 1 .1 .8 -.1 .2

(I-A)^-1
.7 -.4 = .2 .4 ----> ad-bc= .1
-.1 .2 .1 .7 divide everything by .1

(I-A)^-1 x D
2 4 x 20 = 280
1 7 60 440

you need to produce 280 units of coal and 440 units of steel to produce 20 units of coal and 60 units of steel

I-A and then superaugment that to I, once that is simplified multiple it by D

x>=0, y>=0

- pick points on the rays that keep going
- test those along with the other vertices that bind the set
-max and min can't be the test points

use this instead of Bernoulli process when a particular outcome depends on the outcome of the preceding trial

probability that a driver will be downtown after dropping off his SECOND customer?
X2= X0 x P^2

X0= [.2 .8]
P^2= .45 .55
.44 .56
X2= [.442 .558]

determine the probability of a particular outcome of the kth stage of the Markov process

WP=W

add w1+w2=1 to one of the two equations (aka drop one)

P= 1-a a
1-b b

W= b/(a+b) a/(a+b)

WP=W

get all the variables to one side to make it equal 0
add w1+w2=1 to one of the three equations (aka drop one)
solve to get w3, then w2, then w1

after some number of transitions, P^k has all positive entries