April 14, 2024

Table of contents & cheatsheet (PROBABILITY)

Definitions

Sample space the list of all possible outcomes.

Event the outcomes that meet the requirement.

Probability for event A,

probability

Dependent events two events are dependent if the outcome of event A affects the outcome of event B so that the probability is changed.

Independent events two events are independent if the fact that A occurs does not affect the probability of B occurring.

Conditional probability the probability of A, given that B has happened:

Conditional probability

6.1. Single events

sample event
Mutually exclusive
Mutually exclusive

P(A∪B) = P(A) + P(B)

P(A∩B) = 0

Combined events
Combined events
P(A∪B) = P(A) + P(B) − P(A∩B)
P(A∩B) = P(A) + P(B) − P(A∪B)

A∪B (union)

(union)

A∩B (intersect)

(intersect)

If independent: P(A∩B) = P(A) × P(B).

Compliment, A′ where P(A′) = 1 − P(A)

Exhaustive when everything in the sample space is contained in the events

6.2. Multiple events

Probabilities for successive events can be expressed through tree diagrams or a table of outcomes.

Table of outcomes
table of outcomes
Tree diagram
Tree diagram
In general, if you are dealing with a question that asks for the probability of:
  • one event and another, you multiply
  • one event or another, you add

6.3. Distributions

For a distribution by function the domain of X must be defined as ∑P(X = x) = 1.

Expected value  E(X) = ∑xP(X = x)

Binomial distribution X ∼ B(n, p) used in situations with only 2 possible outcomes and lots of trials

binomial distributions

On calculator

  • Binompdf(n,p,r) P(X = r)
  • Binomcdf(n,p,r) P(x ≤ r)

Mean = np

Variance = npq

 

Normal distribution X ∼ N(μ, σ2)

normal distribution

where μ = mean, σ = standard deviation

On calculator:

  • normcdf(lowerbound, upperbound, = μ, σ)
  • invnorm(area, = μ, σ)