Binomial Distribution — NDA Maths

Formulas (6)

• Bernoulli Trials: One Success, One Failure · Failure complements success
  $$q = 1 - p, \qquad p + q = 1$$
• Reading p and the Success Event from the Story · Odds to probability
  $$\text{odds } a : b \ \Longrightarrow\ p = \dfrac{a}{a + b}$$
• The Binomial Probability Formula · Probability of exactly k successes
  $$P(X = k) = \binom{n}{k} p^{k} q^{\,n-k}$$
• At Least One via the Complement · At least one success
  $$P(X \ge 1) = 1 - q^{n}$$
• Cumulative Probabilities: Summing the Tail · At least two successes
  $$P(X \ge 2) = 1 - P(X = 0) - P(X = 1)$$
• The Complementary Count Y = n − X · Swapping successes for failures
  $$X \sim B(n, p) \ \Longrightarrow\ n - X \sim B(n,\, 1 - p)$$

Watch out for (6)

• 'Until the first success' is not binomial
  → When Is It Binomial? The Four Conditions
• 'Thrice as likely' is 3 : 1, not p = 3
  → Reading p and the Success Event from the Story
• Match the exponents to the success/failure counts
  → The Binomial Probability Formula
• 'At most' can also flip to a complement
  → At Least One via the Complement
• Count the terms before you sum
  → Cumulative Probabilities: Summing the Tail
• n stays the same — only p flips
  → The Complementary Count Y = n − X

Formulas (7)

• Why the Mean Is np and the Variance npq · Mean and variance of one trial
  $$E(I) = p, \qquad \operatorname{Var}(I) = pq$$
• Mean, Variance, and Standard Deviation · The three summary measures
  $$\mu = np, \qquad \sigma^2 = npq, \qquad \sigma = \sqrt{npq}$$
• Recovering n and p from the Moments · Divide variance by mean to get q
  $$q = \dfrac{\sigma^2}{\mu} = \dfrac{npq}{np}$$
• When You Are Given a Relation, Not the Values · Mean equals c times variance
  $$np = c\,(npq) \ \Longrightarrow\ q = \dfrac{1}{c}$$
• Finding p from a Probability Equation · Ratio of two probabilities
  $$\dfrac{P(X=b)}{P(X=a)} = \dfrac{\binom{n}{b}}{\binom{n}{a}}\, p^{\,b-a}\, q^{\,a-b}$$
• Variance Is Unchanged by Y = n − X · Variance survives the swap
  $$\operatorname{Var}(n - X) = npq = \operatorname{Var}(X)$$
• The Symmetric Case: Mean = n/2 when p = ½ · Symmetric binomial mean
  $$p = \tfrac12 \ \Longrightarrow\ \mu = \dfrac{n}{2}$$

Watch out for (6)

• Standard deviation is √(npq), not npq
  → Mean, Variance, and Standard Deviation
• Variance over mean gives q, not p
  → Recovering n and p from the Moments
• Cancel np, do not cancel the wrong factor
  → When You Are Given a Relation, Not the Values
• Use coefficient symmetry before brute force
  → Finding p from a Probability Equation
• Variance does not flip; the mean does
  → Variance Is Unchanged by Y = n − X
• Symmetry needs p = ½, not just 'two outcomes'
  → The Symmetric Case: Mean = n/2 when p = ½