1.1: Review of Probability Theory Flashcards
(29 cards)
Random Experiment
process producing one of many unpredictable outcomes
Variation
the variability in the outcomes of random experiment
Sample Space,
Notation?
list of all possible outcomes in random experiment
Notation: S
What does n(S) mean
number of outcomes in the sample space
Event
collection of outcomes that share some common attribute, basically a subset in the sample space
What is n(A):
the number of times an Event occurs
Probability
expeced frequency of times the event will occur when random experiment is repeated 1000000000 times
What are the conditions of probability in a random experiment?
- probability of all outcomes should be >= 0
- Sum of all the probabilities of outcomes = 1
- For an event A, The probability of it occuring is -> number of times A occurs/number of outcomes in sample space
P(a) = n(A)/n(S)
Mathematical Probability
Frequentist Probability
M: list it, count it, get the answer, n(A)/n(S)
F: write the code, run it, and evaluate (this is simulation)
Example of Frequentist probability vs Mathematical probability
P(of rolling a 6), and in 10 rolls you roll it 1 time on the third role
MP: P(rolling a 6)= 1/6
FP: 0/1, 0/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8….
If you roll it 100,000 times eventually it will become 1/6
Law of Large Numbers
process of repeating a random experiment many timeks over to compute probability of an event when sample space is unknown
this is why the frequentist probability eventually equals the mathematic probability
What is A∪B
Set that consists of any elements found in:
A
B
or both
What is A ∩ B
Area overlapping A & B
- football area
What is Addition Law
To Calculate P(A∪B):
P(A∪B) = P(A) + P(B)- P(A∩B)
Everything= All of A + All of B - Overlap
What is Mutually Exclusive
When P(A∩B) = 0
no overlap, it is empty, or null set
What is A^c
Complement is the subset of all outcomes in Sample space that are not in event A
Everything but A
A, A^c, S
Event A, ANYTHING BUT A, and everything total
How to calculate P(A^c)
probability of A complement, OR ALTERNATIVELY TITLED: Law of Complement
P(A^c)= 1-P(A)
Demorgan Laws
- P(A∪B)^c = P(A^c ∩ B^c)
- P(A∩B)^c = P(A^c ∪ B^c)
Multiplicaive rule
If you have m trials, each with r outcomes, you can calculate the sample space as
r^m
For example, how many 3 digit numbers are there?
m=3
r= 10
10^3= 1000
Permutation
ORDERED ARRANGEMENTS of n distinct objects
n! = n(n-1)(n-2)
R code factorial?
factorial(52)
Permutation Rule
What are the number of ways to permutate R distinct objects chosen from n distinct objects? arrange R distinct objects from n sample space
nPr = n!/ (n-r)!
Combination
when order does not matter
(n r) = n!/(r!(n-r)!)
