A single yes/no trial β the atom every other discrete distribution is built from. $E[X]=p$, $\mathrm{Var}(X)=p(1-p)$, maxed at $p=\tfrac12$.
Method · Bernoulli
Intro
Bernoulli is the foundational atom: a single yes/no trial. The next distribution (Binomial) answers 'what if I run n independent Bernoullis and count the successes?'
β Intro Β· expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine β being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
β Try first Β· expand
Worked example
A biased coin lands heads with probability p = 0.3. Let X = 1 if the coin lands heads on a single toss, 0 otherwise. (a) Identify the distribution of X. (b) Compute E[X] and Var(X). (c) Compute P(X = 1) and P(X = 0).
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
Where does this distribution sit in the story chain β what question does it answer that the previous distribution couldn't? Try to recall the key moment formulas (mean, variance) without looking them up.