### Deterministic possibilities
All the different possible outcomes.
`Random two bit number has deterministic possibilities 00, 01, 10, 11.`

### Probability distribution
A Probability for each possibility: Pr\[x] = p where p $\in$ \[0,1]
`For a coin flip this would be Pr[Heads] = 1/2 and Pr[Tails] = 1/2`
Combine all probabilities for all possible outcomes as vector to get probability distribution+
A valid distribution d has $\sum d_i = 1$ and for all i is $d_i \geq 0$ 
For a coin flip the probability distribution would be $d_coin \in \mathbb{R}^2$ with $d = (1/2, 1/2)^t$ 

### Probabilistic processes
A probabilistic process is a Matrix A $\in$ $\mathbb{R}^{N \times N}$ if every column of A is a valid probability distribution.
![[Pasted image 20260728163713.png]]
An example for generating a 2 bit number and then using a bit flip device to flip both 
bits with a probability of 1/3 
For two zeroes this results in 1/3 chance to get 11 and 2/3 chance for nothing to happen.
Here, A would look like this:
![[Pasted image 20260728164547.png]]

### Applying a probabilistic process
A Process A applied to a distribution x is defined as Ax.
![[Pasted image 20260728164820.png]]