QM

Question 1

Single events

Answer 1

one specific
outcome

Question 2

Multiple events

Answer 2

more than
one outcome

Question 3

Union

Answer 3

arba. At least one of A and B occurs
(any element in A or B

Question 4

Intersection

Answer 4

ir. Both A and B occur

Question 5

Subset

Answer 5

If A occurs, then B occurs

Question 6

Disjoint

Answer 6

A and B cannot occur jointly

Question 7

Partition

Answer 7

The sets D1, D2, …, Ds form a partition if they are mutually disjoint and their union is Omega (entire sample space)

Question 8

Classical definition of probability

Answer 8

1. P(O/) = 0 and P(O/) = 1
2. 0 ≤ P(A) ≤ 1 for all events A: 0/𝑁≤𝑁𝐴/𝑁≤𝑁/𝑁
3. P(A union B) = P(A) + P(B) if A, B disjoint

all outcomes of an experiment are equally likely

Question 9

arbitrarily

Answer 9

= randomly

Question 10

Empirical Definition of Probability

Answer 10

Requirement: the random experiment is independently and identically
repeatable.
In n trials, n(A) is the number of trials where A occurs. The law of large numbers states that:
1) The ratio n/n(A) approaches a constant as 𝒏 → ∞
2) This constant is P(A), same properties hold.

probability of an event is determined by its relative frequency over a large number of trials

Question 11

Subjective Definition of Probability

Answer 11

Probability P(A) reflects how strongly an individual believes in the future occurrence of event A.
* Can be used for all random experiments
* Disadvantage: it is subjective – different people may obtain different
probabilities for the same event
* Same properties should hold

Question 12

General Definition of Kolmogorov: basic axioms of a probability model

Answer 12

A probability measure or model P assigns
real numbers P(A) to all events A (subsets of Ω), in such a way that:
(1) P(A) >= 0
(2) P() = 1
(3) If A, B disjoint, then P(AB) = P(A) + P(B)

All three definitions discussed above provide a probability model.

Question 13

Random drawing

Answer 13

one population element is arbitrarily chosen

Question 14

Random sampling

Answer 14

randomly drawing n elements from the population

Question 15

In how many ways can we order k objects?
Number of orderings of k objects.

Answer 15

k!

Question 16

In how many ways can k objects be chosen from a set of m objects?
Number of possibilities to choose k objects from m objects.

Answer 16

k-tuple: chosen sequence of k objects.

a) ordered, with replacement (=after each draw the chosen object is put back
into the population)
b) ordered, without replacement
c) unordered, without replacement
d) unordered, with replacement (not considered here)

Question 17

(2a) ordered, with replacement (k objects from the set of m objects)

Answer 17

• m possibilities to choose for the first position
• the first drawn object is replaced back to set m -> again, m possibilities
  to choose for the second position, etc
• number of possibilities for all k choices equal to m
  Total number of orderings: 𝑚^k

Question 18

(2b) ordered, without replacement (k objects from the set of m objects)

Answer 18

• m possibilities to choose for the first position
• m-1 possibility to choose for the second, m-2 for the third, etc
• m-k+1 possibilities at the last, kth choice

𝑚!/(𝑚−𝑘)!

Question 19

(2c) unordered, without replacement (k objects from the set of m objects)

Answer 19

• Compared to 2b), groups of k! different orderings become equal
• Need to take the number of orderings from 2b and divide it by k! to get the
  unordered number pf k-tuples without replacement
  Total number of orderings: 𝑚!
  𝑘!×(𝑚−𝑘)!= (𝑚
  𝑘) (taip ir atrodo)
• Pronounced “m choose k”
• Called binomial coefficients

Question 20

Conditional probability

Answer 20

Conditional probability of event A given that event B occurs:
𝑃𝐴|𝐵 =𝑃𝐴∩𝐵 / 𝑃(𝐵)

Pronunciation: probability of A given B
Another notation: PB(A)
In this context : P(A) is the prior probability
(before the info B became available)
P(A|B) is the posterior probability
(after the info B became available)

Question 21

Conditional probability: Rules

Answer 21

Product rule (Notion WEEK 2 (1 ft))

Question 22

A and B are independent if

Answer 22

if 𝑃𝐴𝐵 =𝑃(A)

~ if the pre-information about the occurrence of B doesn’t influence the probability of A

Question 23

Bayes’s Rule

Answer 23

Notion WEEK 2

Question 24

Sample

Answer 24

Some subset of a population

Question 25

Population

Answer 25

The entirety

Question 26

Random variable (RV)

Answer 26

function that takes outcome of experiment and returns a measure/value