Q.If $P(X=0)=1-P(X=1)$ and $E(X)=3\,\text{Var}(X)$, then $P(X=0)$ is

A Bernoulli trial is a random experiment with exactly two mutually exclusive, exhaustive outcomes — success or failure — repeated independently with the success probability $p$ held fixed across repetitions.

Bernoulli distribution (Definition 11.10): with $X(\text{success})=1$, $X(\text{failure})=0$,

Mean $=p$, variance $=pq$ (derived in the Mean-&-Variance-of-Standard-Distributions concept).

Binomial distribution (Definitions 11.11-11.12): if $X$ counts the number of successes across $n$ independent Bernoulli trials, each with the same success probability $p$, then $X\sim B(n,p)$ with pmf

$\binom nx$ counts the number of ways to choose which $x$ of the $n$ trials were successes; each such arrangement has probability $p^xq^{n-x}$ by independence, and summing $\binom nx p^xq^{n-x}$ over $x=0,\dots,n$ recovers the binomial expansion $(p+q)^n=1^n=1$ — the reason for the name "binomial". …