Definitions
Sample space the list of all possible outcomes.
Event the outcomes that meet the requirement.
Probability for event A,
![probability](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-25-at-10.40.15-AM.png)
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](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-25-at-10.41.34-AM.png)
6.1. Single events
![sample event](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-09-46-04-88_e2d5b3f32b79de1d45acd1fad96fbb0f-768x276.jpg)
![Mutually exclusive](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-09-44-40-43_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg)
P(A∪B) = P(A) + P(B)
P(A∩B) = 0
![Combined events](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-09-46-28-07_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg)
A∪B (union)
![(union)](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-09-46-52-56_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg)
A∩B (intersect)
![(intersect)](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-09-47-11-01_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg)
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](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-10-42-15-13_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg)
![Tree diagram](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-09-58-29-95_e2d5b3f32b79de1d45acd1fad96fbb0f-300x172.jpg)
- 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](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot-2023-03-25-at-12.03.59-PM.png)
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](https://revisiontown.com/wp-content/uploads/2023/03/Screenshot_2023-03-25-10-42-52-85_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg)
where μ = mean, σ = standard deviation
On calculator:
- normcdf(lowerbound, upperbound, = μ, σ)
- invnorm(area, = μ, σ)