Trial vs realization In the context of random variables, is "trial" synonymous with "realization"? For example, is a Bernoulli trial one realization of a Bernoulli random variable?
 A: An example of Bernoulli trial is flipping a coin (see https://en.m.wikipedia.org/wiki/Bernoulli_trial). 
Before the coin is flipped, you can define an observable random variable $X$ which will keep track of the outcome of the coin flip (i.e., heads or tails). Specifically:
$X = 1$ if the coin flip will land heads and 
   $X = 0$ if the coin flip will land heads. 
The reason the random variable $X$ is observable is because its value will become known to us after flipping the coin.  As remarked by @whuber, there are also unobservable random variables, which will not be a concern to us in this answer. The values of unobservable random variables will remain unknown to us even after conducting a random experiment such as a coin flip. 
As pointed out by @whuber in his answer, the observable random variable $X$ will "keep track of the experimental outcomes associated with a Bernoulli trial by assigning numbers to them in systematic, consistent ways".  Furthermore, assigning numbers to these experimental outcomes facilitates posing and answering quantitative questions involving the random variable $X$.    
You won't know with certainty what the value of $X$ is before you perform a single flip - in other words, you are uncertain about whether the coin flip will land heads (in which case the value of $X$ would be equal to $1$) or tails (in which case the value of $X$ would be equal to $0$). However, if you were to perform a large number of coin flips with a fair coin, you would expect half of those flips to land heads and half to land tails. In other words, you would expect the probability that $X = 1$ to be 1/2 and the probability that $X = 0$ to be 1/2.
After the coin is flipped, you will observe the actual value of the observable random variable $X$. This value is called a realization of $X$. For example, the realized value of $X$ may be 0 (if the coin flip produced tails). Note that this value is now known with certainty. The uncertainty that was in place before conducting the coin flip with regards to possible value of $X$ was replaced with certainty about the actual value of $X$ observed after conducting the coin flip. 
So many people struggle with the concept of random variable because they don't understand that:


*

*A random variable keeps track of the outcome of a random experiment (e.g., flipping a fair coin);

*The value of the random variable is unknown only before conducting the random experiment;

*After conducting the experiment, the value of the random variable is fully known if the variable is observable.

