For treatment assignment, what is the difference between Bernoulli assignment vs. completely randomized assignment? I have read in literature several times a distinction being made between Bernoulli assignments and completely randomized assignments in assigning treatment (1) vs control (0) to units in a study. I was wondering if anyone could point me in a direction to distinguish between the two. What is the difference between the two? Is there a connection to multinomial sampling? Thanks.
 A: Let's imagine you have a group of $n$ people and you want to separate them between treatment and control groups. 
Bernoulli trials
In a bernoulli assignment, you consider each person individually and "flip" a coin with probability $p$ for assigning the treatment to that person.
On average, you will have $np$ persons in the treatment group, but you could have very unbalanced groups by chance! 
For example, suppose you have $n = 20$ and each has $p = 50\%$ chance of getting treatment. On average, you would get $np = 10$ treated units. In practice, you could get a very unbalanced sample. 
Let's see this by simulating 1000 different bernoulli assignments in R:
rm(list = ls())
set.seed(10)
n <- 20
hist(replicate(1000, sum(rbinom(n, 1, 0.5))), 
     main = "bernoulli assignment",
     xlab = "number of treated",
     col = "lightblue")


Notice how often you get a very unequal distribution between treated and control groups. You could even have the misfortune of having zero treated or zero control units, which wouldn't allow you to compare anything.
This randomization scheme has $2^n$ possible outcomes, in our example $2^{20} = 1,048,576$, that is, more than 1 million different possible treatment assignments vectors for just these group of 20 people. This can be harmful to the precision of your inference.
Completely randomized experiment
But there's a way to keep the probability of an individual being treated equal to $p$ while making sure you get a specified number of treated units. 
Let's go back to our case of $n = 20$ and suppose you want to make sure $10$ of them get treated and all of them have equal chance of being treated ($p = 0.5$). 
To do that, imagine labeling each individual from  1 to 20, putting those labels on a box, and then taking a random sample of 10 labels from the box. In R you could do:
sample(1:20, 10)

Then this guarantees you will always have the sample size you want while maintaining the probability of each individual getting the treatment the same.
Comparing to the bernoulli trials, notice this randomization scheme has only
$$n \choose \frac{n}{2}$$ 
possible outcomes, in our example, only 184,756 possible treatment assignment vectors, much less than 1 million. Intuitively, you can think that you are eliminating the uninformative samples that the bernoulli trials would give you.
