Introduction to Probability – Blitzstein & Hwang (Ch1–Ch6.3)
This summary is based on the book, but also on the YouTube videos from Harvard based on the
book. The first 17 lectures are taken into account from the following playlist:
https://www.youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6ht66KWxbzTIo
Chapter 1: Probability and counting
Naive probability assumes a finite sample space where every possible outcome were equally
likely.
P naive( A)=1 – P naive ¿).
Binomial coefficient (denoted as (nk )) counts the possible ways to choose a subset of size k set of
size n for k≤n. It is not possible if k>n.
n
Binomial theorem: ( x + y ) = ∑ x y
n k n−k
k=0
We need to choose k times from a set of n objects:
- Without replacement and order does not matter: (nk) possible ways to choose
- Without replacement and order does matter: n ×(n−1) ×... ×( n−k +1) possible ways
- With replacement and order matters: n k possible ways
- With replacement and order does not matter: (n+ k−1
k )
possible ways.
Axioms of Probability:
- P( Ø )=0 , P( S)=1
- If A1, A2, ..., An are disjoint then the probability of the union of all As is equal to the sum of the
probabilities of each A.
Properties of Probability:
- P( A c )=1 – P( A)
- If A C B, then P( A) ≤ P(B)
- P( A U B)=P( A)+ P(B) – P( A ∩ B) (=Inclusion-exclusion principle). Sometimes
P( A ∩ B)=Ø , then they are disjoint and you can leave that part out, but do include "since
P( A ∩ B)=Ø " or add that they are disjoint.
Pigeonhole principle: if you have more dots than boxes, there needs to be a box with more than
one dot.
A Venn diagram illustrates events as ovals and intersections of event as corresponding parts of the
ovals.
Chapter 2: Conditional probability
This summary is based on the book, but also on the YouTube videos from Harvard based on the
book. The first 17 lectures are taken into account from the following playlist:
https://www.youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6ht66KWxbzTIo
Chapter 1: Probability and counting
Naive probability assumes a finite sample space where every possible outcome were equally
likely.
P naive( A)=1 – P naive ¿).
Binomial coefficient (denoted as (nk )) counts the possible ways to choose a subset of size k set of
size n for k≤n. It is not possible if k>n.
n
Binomial theorem: ( x + y ) = ∑ x y
n k n−k
k=0
We need to choose k times from a set of n objects:
- Without replacement and order does not matter: (nk) possible ways to choose
- Without replacement and order does matter: n ×(n−1) ×... ×( n−k +1) possible ways
- With replacement and order matters: n k possible ways
- With replacement and order does not matter: (n+ k−1
k )
possible ways.
Axioms of Probability:
- P( Ø )=0 , P( S)=1
- If A1, A2, ..., An are disjoint then the probability of the union of all As is equal to the sum of the
probabilities of each A.
Properties of Probability:
- P( A c )=1 – P( A)
- If A C B, then P( A) ≤ P(B)
- P( A U B)=P( A)+ P(B) – P( A ∩ B) (=Inclusion-exclusion principle). Sometimes
P( A ∩ B)=Ø , then they are disjoint and you can leave that part out, but do include "since
P( A ∩ B)=Ø " or add that they are disjoint.
Pigeonhole principle: if you have more dots than boxes, there needs to be a box with more than
one dot.
A Venn diagram illustrates events as ovals and intersections of event as corresponding parts of the
ovals.
Chapter 2: Conditional probability
