Chapter Overview
🧭Your visual roadmap through the chapter
Binomial
Fixed trials, 2 outcomes, constant probability
Hyper-geometric
Sampling without replacement
Poisson
Events at a constant average rate
Normal
Continuous, bell-shaped; ± approximations
Theoretical probability distributions used to estimate the probability of a given result when a sample is taken from a population with known characteristics.
Choosing and applying the right distribution is central to quality control, risk estimation and forecasting problems in business.
Recognise which distribution applies; calculate probabilities using each formula; use Poisson/Normal tables; apply the normal approximation with continuity correction.
Key Definitions
📘Source-based terminology, exam-ready
Probability Distribution
Similar to a frequency distribution but using probabilities instead of frequencies; the total always equals 1.
Key word: totals to 1
Probability Function
A formula giving the probability of a random variable taking a specific value.
Key word: formula for P(x)
Theoretical vs Experimental
Theoretical distributions give expected probability when outcomes are equally likely; experimental distributions result from actual observed occurrences.
Key word: expected vs observed
Discrete vs Continuous Distribution
Discrete distributions record a finite number of outcomes; continuous distributions record an infinite number of possibilities (e.g. time, distance).
Key word: finite vs infinite
Binomial Distribution
Applies to a discrete variable with a fixed number of independent trials, each with two outcomes and a known, constant probability of success.
Key word: fixed trials, constant p
Hyper-geometric Distribution
Applies when a sample is randomly selected WITHOUT replacement from a population with two types of items; trials are dependent.
Key word: no replacement
Poisson Distribution
A discrete distribution for events that occur randomly, independently, singly, and at a constant average rate, over a period of time or space.
Key word: rate, not probability
Normal Distribution & Z-score
A symmetric, bell-shaped continuous distribution. The Z-score (standardized normal variate) measures how many standard deviations a value is from the mean.
Key word: Z=(x−μ)/σ
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Discrete Distributions
MEANING
Binomial, Hyper-geometric and Poisson each model a countable number of "successes" under different conditions.
PURPOSE
Estimate the probability of a specific count of outcomes.
APPLICATION
Quality control (defect counts), sampling problems, event-rate forecasting.
CONCEPT → Normal Distribution
MEANING
A continuous, symmetric bell-shaped distribution fitting many naturally occurring variables.
PURPOSE
Estimate the probability of a value falling in a given range using Z-scores and tables.
APPLICATION
Heights, weights, product dimensions; also approximates Binomial/Poisson when conditions are met.
Key Rules & Principles
📏Conditions for each distribution to apply
RULE 1 Binomial — 4 conditions
Fixed number of trials; each trial independent
Each trial has exactly two outcomes (success/failure)
Probability of success is known and CONSTANT for every trial (implies replacement, or a population so large the effect is negligible)
RULE 3 Poisson — 4 conditions
Events occur randomly, independently, and singly (one at a time)
Events happen, on average, at a constant rate (λ)
Models HOW OFTEN something happens, not the probability of one trial
RULE 4 Normal distribution & approximations
Normal curve: symmetric, bell-shaped; mean = median = mode; total area = 1
Normal approximates Binomial when 0.1<p<0.9 and np>5
Normal approximates Poisson when λ > 30
Poisson can approximate Binomial when n is large and p is small
A continuity correction (±0.5) is required whenever a continuous distribution approximates a discrete one
RULE 2 Hyper-geometric — 4 conditions
Outcomes classified success/failure; sample taken WITHOUT replacement
Probability of success CHANGES each trial — trials are dependent
The experiment is repeated a definite number of times
Formula Zone — Discrete Distributions
🧮Every formula supported by the source text
BINOMIAL DISTRIBUTION
p=P(success), q=P(failure)=1−p, n=sample size. Mean=np; SD=√(npq).
HYPER-GEOMETRIC DISTRIBUTION
N=population size, n=sample size, k=possible successes in population, x=successes in sample.
POISSON DISTRIBUTION
λ=average rate; e=2.71828. Mean=λ=np; SD=√λ.
POISSON RECURSIVE SHORTCUT
Each successive probability = previous probability × λ/n. P(x=0) is always e−λ.
Choosing Between "or fewer" and "at least" (Poisson & Binomial ranges)
"x or fewer": sum the individual probabilities of 0 up to x (or read directly from a cumulative table).
"at least x" (= x or more): since all probabilities sum to 1, calculate 1 − P(fewer than x) instead of summing an open-ended list.
Formula Zone — Normal Distribution
🧮Every formula supported by the source text
Z-SCORE (STANDARDIZED NORMAL VARIATE)
x=value under investigation, μ=mean, σ=standard deviation. Converts any normal distribution to the standard one (mean 0, SD 1).
NORMAL APPROXIMATION TO BINOMIAL
Valid when 0.1<p<0.9 and np>5. Requires continuity correction.
NORMAL APPROXIMATION TO POISSON
Valid when λ > 30. Requires continuity correction.
KEY AREAS UNDER THE NORMAL CURVE
Also: ±1.96σ encloses 95%; ±2.58σ encloses 99% of the total distribution.
Continuity Correction Table (discrete event → corrected normal range)
| Discrete event | Normal-approximation correction |
|---|---|
| Exactly 10 successes: P(x=10) | P(9.5 ≤ x ≤ 10.5) |
| More than 10: P(x>10) | P(x > 10.5) |
| 10 or more: P(x≥10) | P(x > 9.5) |
| Fewer than 10: P(x<10) | P(x < 9.5) |
| 10 or fewer: P(x≤10) | P(x < 10.5) |
Calculation Roadmap
🗺️General path for any distribution question
1· Read Question
Note trials, replacement, and what's asked
2· Identify Distribution
Binomial, Hyper-geometric, Poisson, or Normal?
3· Select Formula
Direct calc, table lookup, or approximation
4· Substitute
Put known values into the formula
5· Calculate & Check
Solve, recheck, state final answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Binomial Probability
- 1Confirm the 4 binomial conditions are met; identify n, p, q, x
- 2Calculate the binomial coefficient n!÷[x!(n−x)!]
METHOD D · Normal Distribution (Find a Probability)
- 1Identify the mean (μ), standard deviation (σ), and value(s) x
- 2Calculate Z = (x−μ)÷σ for each value
- 3Look up the Z value in the normal table (area for half the curve)
- 4Add/subtract 0.5 or combine areas depending on what's asked (above/below/between)
3 Multiply by px and qn−x (p to the x, q to the n−x)
- 4For a range, sum individual P(x) values or use the complement
METHOD B · Hyper-geometric Probability
- 1Identify N (population), n (sample), k (successes in population), x (successes wanted)
- 2Calculate kCx and (N−k)C(n−x) separately
- 3Multiply those two together
- 4Divide by NCn
METHOD E · Normal Distribution (Find a Value from %)
- 1Identify the target proportion/percentile
- 2Look up the matching Z value in the normal table (reverse lookup)
- 3Rearrange Z=(x−μ)/σ to x = μ ± Zσ
- 4Substitute μ, σ and Z to find x
METHOD C · Poisson Probability
- 1Identify λ (average rate) for the period/space asked about — rescale if needed
METHOD F · Normal Approximation (Binomial or Poisson)
- 1Check the approximation condition is met (0.1<p<0.9 & np>5, or λ>30)
- 2Calculate the approximating mean and SD
- 3Apply the continuity correction (±0.5) to match the wording
- 4Calculate Z and read off the probability as in Method D
2 Calculate e−λ once
3 Substitute into P(x)=e−λλx/x!, or use the recursive shortcut for consecutive x values
- 4For "at least", use 1−P(fewer)
Different Cases / Scenarios
🔀Recognise the situation, pick the right distribution
IF trials are fixed, independent, WITH replacement (or a huge population)
→ USE the Binomial distribution
IF sampling is WITHOUT replacement from a small/finite population
→ USE the Hyper-geometric distribution
IF you're counting random events over time/space at a constant rate
→ USE the Poisson distribution
IF the variable is continuous and roughly bell-shaped/symmetric
→ USE the Normal distribution
IF n is large and p is small (Binomial hard to compute directly)
→ Approximate with Poisson
IF n is large and 0.1<p<0.9 with np>5
→ Approximate the Binomial with the Normal (+ continuity correction)
IF λ (Poisson rate) is greater than 30
→ Approximate the Poisson with the Normal (+ continuity correction)
IF asked for "at least"/"more than" x
→ USE the complement: 1 − P(fewer than/up to x)
Comparison Tables
⚖️Don't confuse these!
| Binomial | Hyper-geometric | Poisson | |
|---|---|---|---|
| Replacement | With (or negligible effect) | Without | N/A — models a rate |
| Trials | Independent, fixed n | Dependent, fixed n | Independent, over time/space |
| Probability of success | Constant (p) | Changes each trial | Governed by constant rate λ |
| Discrete Distributions (Binomial/Hyper-geo/Poisson) | Normal Distribution | |
|---|---|---|
| Variable type | Discrete (countable) | Continuous |
| Output | Exact P(x) for a whole-number count | Area/probability over a range, via Z-score |
| Link between them | — | Normal can approximate Binomial/Poisson (with continuity correction) |
| Normal Approximation | Condition | Mean & SD Used |
|---|---|---|
| To Binomial | 0.1 < p < 0.9 and np > 5 | Mean=np, SD=√[np(1−p)] |
| To Poisson | λ > 30 | Mean=λ, SD=√λ |
Reading Tables & the Normal Curve
📌How to interpret the source's tools
Reading a Poisson table
Columns are values of λ; rows are values of x. The cell gives P(x) directly for that combination — no calculation needed. A cumulative Poisson table instead gives P(x or fewer).
Reading the Normal (Z) table
The table gives the area between the mean and Z standard deviations — i.e. HALF the distribution only. Combine areas (add or subtract 0.5) depending on whether the question asks above/below/between values.
Reading the normal curve shape
A higher standard deviation gives a flatter, more spread-out curve; a lower standard deviation gives a taller, narrower curve. All normal curves are symmetric about the mean regardless of shape.
Combining Z areas for a range
For a value between two points straddling the mean, ADD the two individual Z areas. For two points on the same side of the mean, SUBTRACT the smaller Z area from the larger.
Worked Calculation Logic
✍️Method students must reproduce in the exam
BINOMIAL: GIVEN → FORMULA → SUBSTITUTION → ANSWER
10% of homes have an occupant aged 90+. What is the probability that a sample of 10 homes contains exactly 2 such homes?
GIVEN: n=10, p=0.1, q=0.9, x=2
FORMULA: P(x) = [n!/(x!(n−x)!)] pxqn−x
SUBSTITUTION: P(2) = [10!/(2!8!)] × 0.1² × 0.9⁸=45×0.01×0.4305
ANSWER: P(2) = 0.1937 or 19.37%
NORMAL: GIVEN → FORMULA → SUBSTITUTION → ANSWER
Bags of sugar: mean=1,010g, SD=5g. What's the probability a bag weighs less than 1,000g?
GIVEN: x=1,000, μ=1,010, σ=5
FORMULA: Z=(x−μ)/σ
SUBSTITUTION: Z=(1,000−1,010)/5 = −2. Table: area for Z=2 is 0.4772
ANSWER: P(x<1,000) = 0.5−0.4772 = 0.0228 or 2.28%
Exam Question Patterns
📝Identification — which distribution applies to a described scenario
Direct calculation — Binomial, Hyper-geometric or Poisson P(x)
Table reading — Poisson or Normal Z tables
Normal probability — using Z-scores for a range
Reverse lookup — finding x from a given percentage
Normal approximation with continuity correction
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Wrong distribution choice — check replacement (Binomial) vs no replacement (Hyper- geometric) carefully.
- ✗ Forgetting the continuity correction when using Normal to approximate a discrete distribution.
- ✗ Reading the Z table as the full probability — it only gives HALF the distribution's area.
- ✗ Sign/direction errors — a negative Z means below the mean; check which side the question needs.
- ✗ "At least" vs "more than" — these use different correction boundaries (9.5 vs 10.5); check the continuity correction table.
Quick Decision Tree
🌳Chapter 12 in 60 Seconds
⚡Binomial: replacement, constant p, mean=np
Hyper-geo: no replacement, dependent trials
Poisson: constant rate λ, mean=λ
Normal: Z=(x−μ)/σ, table=half curve
Approx Binomial→Poisson: n large, p small
Approx →Normal: np>5 or λ>30
Continuity correction: ±0.5 "At least x" = 1−P(fewer than x)
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