The binomial distribution arises from n independent Bernoulli trials with constant success probability p.

Conditions for Binomial:

1. Fixed number of trials n
2. Each trial is independent
3. Each trial has exactly two outcomes
4. P(success) = p is constant across trials

Key formulas:

• P(X=r) = C(n,r)*$p^r$*q^(n-r), where q = 1-p
• E(X) = np (average number of successes)
• Var(X) = npq (spread of the distribution)
• Mode = floor((n+1)p) when (n+1)p is not an integer

Finding parameters from conditions:

• Given mean and variance: q = $\frac{Var}{Mean}$, p = 1-q, n = $\frac{Mean}{p}$
• Example: mean=4, var=3 gives q=3/4, p=1/4, n=16

Common computations:

• P(at least k): use complement for small k. P(X>=2) = 1-P(0)-P(1)
• P(at most k): direct sum P(0)+P(1)+...+P(k)
• P(exactly k): single term C(n,k)*$p^k$*q^(n-k)

Ratio test for mode: P$\frac{X=r}{P}$(X=r-1) = (n-r+1)$\frac{p}{rq}$. P(X=r) increases while this ratio > 1.

When NOT to use binomial:

• Drawing without replacement (use hypergeometric)
• Variable number of trials (use geometric)
• More than two outcomes per trial (use multinomial)