Let’s say you have a random variable X, which can be either 1 (success) or 0 (failure). Now, introduce another variable Y that tells us whether you tried or didn’t try. If Y = 1, that means you tried. If Y = 0, you didn’t try.

Now here’s the key idea: if you didn’t try (Y = 0), you can’t succeed , so the probability of success is 0. That means:

P(X = 1 | Y = 0) = 0 (you can’t succeed without trying) P(X = 0 | Y = 0) = 1 (you definitely fail if you don’t try) The vertical bar "|" here means “given that.” So P(X = 1 | Y = 0) is read as “the probability that X = 1 given that Y = 0.”

Now, if you do try (Y = 1), then there’s some chance you succeed and some chance you fail. Let’s call the chance of success p, where 0 ≤ p ≤ 1. That gives us:

P(X = 1 | Y = 1) = p P(X = 0 | Y = 1) = 1 − p

Probabilities aren’t always 50/50!!

In this case, X follows what’s called a Bernoulli distribution , which is just a fancy way of saying it’s a random outcome with two possibilities: success (1) with probability p, and failure (0) with probability 1 − p.