Binomial and Poisson Distributions
FoundationThe binomial distribution counts successes in a fixed number of independent trials, with mean np; the Poisson distribution counts random events per unit of time, with mean and variance both equal to the rate λ. Read the context, choose the right distribution, and these problems follow fixed routines.
- A discrete random variable's probability distribution lists each value's probability, the total being 1
- Binomial X ~ B(n, p): P(X = r) = C(n,r)p^r(1 − p)^(n − r), E(X) = np, Var(X) = np(1 − p)
- Poisson X ~ Po(λ): P(X = r) = e^(−λ)λ^r/r!, with E(X) = Var(X) = λ
文體小知識：數學概念說明文——由離散隨機變數與伯努利試驗出發，建立二項分佈與泊松分佈的概率公式、期望值與方差，並比較兩種分佈各自的適用情境。
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1Many experiments have just two outcomes — success or failure: tossing a coin, scoring a goal, a product passing inspection. A single such trial is a Bernoulli trial. Repeat the same trial independently n times and the number of successes is a discrete random variable: it takes non-negative integer values with different probabilities, and listing the probability of each value gives its probability distribution, whose probabilities must add up to one.
2Statistics calls this the binomial distribution. Its probability formula has three factors: a combination number, the success probability to the power r, and the failure probability to the remaining power; the combination part counts exactly in which of the trials the r successes occur, and the formula mirrors the binomial theorem. For example, the probability of exactly two heads in five fair tosses is the combination number 10 times one half to the fifth power, i.e. 516.
3For a binomial distribution, the mean is the number of trials times the success probability, and the variance is the number of trials times the success probability times the failure probability — used directly, no proof required. For example, tossing a fair coin a hundred times: the expected number of heads is 50, the variance 25, the standard deviation 5. The mean tells us the average number of successes; the variance measures the fluctuation around it. The closer the success probability is to zero or one, the smaller the fluctuation and the more stable the result.
4Another kind of situation has no fixed number of trials, only an average number of occurrences per unit of time — calls arriving at a service centre each hour, cars passing a junction each minute. The count then follows a Poisson distribution, whose probability formula is: take e to the power minus λ, multiply by λ to the power r, and divide by r factorial, where λ is the average rate and e is about 2.718. The special feature of the Poisson distribution is that its mean and variance are equal, both λ.
5Choose the distribution by context: a fixed number of independent trials with the same success probability each time calls for the binomial distribution; knowing only an average rate per unit of time, with events occurring randomly and independently, calls for the Poisson. For example, the number of sixes in 10 dice throws belongs to the former; an insurer's monthly count of claims belongs to the latter. Both appear side by side in extended-module papers, so the formulas, means and variances must be memorised accurately for instant use.
1Summary 重點摘要
The binomial distribution counts successes in a fixed number of independent trials, with mean np; the Poisson distribution counts random events per unit of time, with mean and variance both equal to the rate λ. Read the context, choose the right distribution, and these problems follow fixed routines.
2Vocabulary cards Vocabulary
A variable that maps the outcomes of a random experiment to numbers, usually denoted by a capital letter.
Taking only finitely many or countably many values; the probabilities of a discrete random variable can be listed one by one.
A table or formula listing each value of a random variable with its probability, all probabilities summing to one.
A single trial with only two outcomes, success and failure, the success probability being p.
The distribution of the number of successes in n independent Bernoulli trials, written X ~ B(n, p).
The distribution of the number of random events per unit of time (or space), written X ~ Po(λ).
The probability-weighted average of a random variable's values — the long-run average result; E(X) = Σ x P(X = x).
A measure of how widely a random variable fluctuates around its mean; both distributions in this lesson have ready-made formulas.
The constant that fixes a distribution's shape: n and p for the binomial, just λ for the Poisson.
The average number of occurrences per unit of time (or space) — the single parameter of the Poisson distribution.
3Step by step Steps
- Step 1: Discrete random variables and probability distributionsA random variable maps experimental outcomes to numbers; one taking only finitely or countably many values is discrete. Describe it with a probability distribution: list each value with its probability, the total being 1. For example, the score of a fair die takes 1 to 6, each with probability 16; the probability of 'at least 5' is P(X = 5) + P(X = 6) = 13. The distribution is the shared framework of both distributions in this lesson: confirm the distribution first, then compute the mean and variance — never reverse the order.
- Step 2: Bernoulli trials and the binomial settingA Bernoulli trial is a single trial with exactly two outcomes, success and failure, the success probability being p. Four conditions for the binomial distribution: the number of trials n is fixed; each trial has two outcomes only; the trials are independent; and the success probability is the same every time. For example, the number of sixes in 10 throws of a fair die: n = 10, p = 16, all four conditions hold, so X ~ B(10, 16). Counter-example: drawing 5 cards without replacement and counting aces — the success probability changes with each draw, so the binomial does not apply and combinations are needed instead. Checking the four conditions before any formula is the first rule of this lesson.
- Step 3: The binomial probability formulaIf X ~ B(n, p), then P(X = r) = C(n,r)p^r(1 − p)^(n − r) for r = 0, 1, 2, …, n. Each factor has a meaning: C(n,r) counts in which of the trials the r successes occur; p^r is the probability of r successes; (1 − p)^(n − r) is the probability that the remaining n − r trials fail. For example, exactly 2 heads in 5 fair tosses: P(X = 2) = C(5,2)(12)²(12)³ = 1032 = 516. For 'at least' or 'at most' questions, add up every qualifying term, or use one minus the complement — never compute just one term.
- Step 4: The binomial mean and varianceIf X ~ B(n, p), then E(X) = np and Var(X) = np(1 − p) — no proof required in the exam, just apply them. For example, a player with a 0.7 shooting rate takes 12 shots: X ~ B(12, 0.7), E(X) = 12 × 0.7 = 8.4, Var(X) = 12 × 0.7 × 0.3 = 2.52, SD ≈ 1.59. When a question asks for the expected number of hits, answer the mean 8.4; only questions about fluctuation involve the variance. Note that as p approaches 0 or 1, the factor (1 − p) or p shrinks, the variance falls and the count becomes more stable — consistent with intuition.
- Step 5: The Poisson distribution and its formulaIf random events occur independently at an average rate λ per unit of time (or space), the count X ~ Po(λ) has P(X = r) = e^(−λ)λ^r/r! for r = 0, 1, 2, … with no upper limit. Here e ≈ 2.718; use the calculator's exponential function. For example, a ticket office receives 3 enquiries per hour on average: X ~ Po(3), and P(X = 2) = e^(−3) × 3²/2! ≈ 0.224. The factorial in the denominator makes large r increasingly unlikely, so counts far above the mean are rare. The exponent of e is always negative — miscopying the sign is the most common slip.
- Step 6: The Poisson mean and varianceThe Poisson distribution's signature is that its mean and variance are both λ: E(X) = λ and Var(X) = λ — a single parameter fixes the whole distribution, unique among the common distributions. For example, a website receives 8 requests per minute on average: X ~ Po(8), mean and variance both 8, SD = √8 ≈ 2.83. In practice, if a data set's mean and variance differ greatly, the Poisson model may not fit. Reading the question also gives signals: 'an average of 3 per hour' — a single rate parameter — hints at Poisson; two numbers, a trial count and a success probability, point to the binomial. Identify before you compute.
- Step 7: Comparing and applying the two distributionsThree questions pick the distribution: is there a fixed number of trials n? Are the trials independent with a common success probability? Are you given n and p, or a rate λ per unit of time? For example, guessing all 10 multiple-choice questions: the number correct X ~ B(10, 14) with E(X) = 2.5; a fast-food shop averaging 12 customers per hour: arrivals Y ~ Po(12) with E(Y) = Var(Y) = 12. For binomial 'at least' questions, add the terms directly; for Poisson ones with many terms, use one minus the complement — e.g. P(X ≥ 3) = 1 − P(X ≤ 2), only three terms — quicker and less error-prone.
5Worked example Example
Example: (a) a fair die is thrown 10 times and X is the number of sixes. State the distribution of X and find P(X ≥ 8). (b) A service centre receives 3 enquiries per hour on average; Y is the number of enquiries per hour and Y ~ Po(3). Find P(Y = 2), and state the mean and variance of Y.
- (a) The probability of a six on each throw is p = 16 and there are 10 independent trials, so X ~ B(10, 16)
- Write the terms: P(X = 8) = C(10,8)(16)⁸(56)², P(X = 9) = C(10,9)(16)⁹(56), P(X = 10) = (16)¹⁰
- Adding the three terms: P(X ≥ 8) = (45 × 25 + 10 × 5 + 1)/6¹⁰ = 117660466176 ≈ 1.94 × 10⁻⁵
- (b) P(Y = 2) = e^(−3) × 3²/2! = (92)e^(−3) ≈ 0.224
- The mean E(Y) = λ = 3 and the variance Var(Y) = λ = 3
- Answer: (a) X ~ B(10, 16) and P(X ≥ 8) ≈ 0.0000194, about two in a hundred thousand — an extremely rare event. (b) P(Y = 2) ≈ 0.224 with E(Y) = Var(Y) = 3. Exam format: state the distribution B(n, p) or Po(λ) first, then the formula, the substitution, and the calculator's value; for 'at least' questions add the terms or use the complement — never a single term.
6Common mistakes Mistakes
7Quick check Quick check
Q1What conditions must hold for the binomial distribution B(n, p)?
Q2Find the probability of exactly 4 heads in 6 fair tosses.
Q3If X ~ B(20, 0.2), find E(X) and Var(X).
Q4If Y ~ Po(2), find P(Y = 0).
Q5In what situation is the Poisson distribution appropriate, and how are its mean and variance related?
8Recap Recap
- A discrete random variable's probability distribution lists each value's probability, the total being 1
- Binomial X ~ B(n, p): P(X = r) = C(n,r)p^r(1 − p)^(n − r), E(X) = np, Var(X) = np(1 − p)
- Poisson X ~ Po(λ): P(X = r) = e^(−λ)λ^r/r!, with E(X) = Var(X) = λ
- Choose by context: fixed-count independent trials take the binomial; a rate per unit time takes the Poisson
- For 'at least' questions, add the terms or use one minus the complement — quick and nothing omitted