What is a Binomial Distribution in the context of Edexcel International A Levels Statistics 2?

A Binomial Distribution is a probability distribution that summarizes the likelihood of a value taking one of two independent states across a series of experiments or trials. In the context of Edexcel International A Levels Statistics 2, it is used to model the number of successes in a fixed number of trials, where each trial has the same probability of success.

How do you identify a Binomial Distribution problem in Mathematics Statistics 2?

To identify a Binomial Distribution problem in Mathematics Statistics 2, check for these conditions: a fixed number of trials, each trial is independent, there are only two possible outcomes (success or failure), and the probability of success is constant for each trial. If these conditions are met, the problem can be modeled using a Binomial Distribution.

What is the formula for calculating probabilities in a Binomial Distribution?

The probability of obtaining exactly k successes in n independent Bernoulli trials is given by the formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the binomial coefficient, p is the probability of success on a single trial, and (1-p) is the probability of failure.

How do you calculate the mean and variance of a Binomial Distribution?

The mean of a Binomial Distribution is calculated as μ = n * p, where n is the number of trials and p is the probability of success. The variance is calculated as σ² = n * p * (1-p). These formulas help summarize the distribution's central tendency and variability.

Can you provide an example of a Binomial Distribution problem relevant to Edexcel International A Levels?

Certainly! Consider a scenario where a student is taking a multiple-choice test with 10 questions, each with four options. If the student guesses on each question, the probability of getting a question right is 0.25. Here, the number of questions (n = 10) and the probability of success (p = 0.25) fit the criteria for a Binomial Distribution. You can use the binomial formula to calculate the probability of getting exactly 5 questions correct.

Why is "Writing $P(X \ge c) = 1 - P(X \le c)$ instead of $1 - P(X \le c - 1)$" a common mistake in Binomial Distributions?

Why it happens: Forgetting that X = c is included in P(X c) but excluded from P(X c - 1). How to avoid it: Always rewrite tail probabilities explicitly: P(X 10) = P(X = 10) + P(X = 11) + = 1 - P(X 9). Source: WST02/01 examiner reports — annual recurring error

Why is "Stating binomial conditions in generic terms without contextual reasoning" a common mistake in Binomial Distributions?

Why it happens: Students memorise 'independent and constant p' but don't apply it. How to avoid it: Always link each assumption to the specific scenario: 'Trials are independent — but in this case students travel together, so independence may fail'. Source: WST02/01 examiner reports

Why is "Confusing variance $npq$ with standard deviation $\sqrt{npq}$" a common mistake in Binomial Distributions?

Why it happens: Both formulae are similar; students forget to take the square root for SD. How to avoid it: Write 'Var = np(1-p)' first, then 'SD = √(Var)' as a second line.

Why is "Rounding $p$, $\binom{n}{r}$ or intermediate products too early" a common mistake in Binomial Distributions?

Why it happens: Calculator displays are limited. How to avoid it: Keep at least 5 significant figures in intermediate working; round only the final answer to 4 sf (or as the question asks).

Why is "Reading $P(X = 4)$ as $P(X \le 4)$ from tables" a common mistake in Binomial Distributions?

Why it happens: Edexcel tables list cumulative values; students miss the column heading. How to avoid it: Use the PMF or BinomialPD for exact P(X = r); use the cumulative table (or BinomialCD) for P(X r). Single point: P(X = r) = P(X r) - P(X r - 1).

Statistics 2Edexcel International A Levels Mathematics (XMA01-YMA01)

Binomial Distributions

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Binomial Distributions Study Notes — Edexcel IAL Mathematics S2 (WST02/01, 2018 spec — 2026 onwards)

The binomial $X \sim B(n, p)$ counts successes in $n$ independent trials with constant success probability $p$ . PMF $\binom{n}{r} p^{r}(1-p)^{n-r}$ , mean $np$ , variance $np(1-p)$ . S2 emphasises cumulative probabilities, tail probabilities (with the off-by-one trap), inverse-cumulative critical values, and modelling judgements about when binomial assumptions are valid.

What you’ll learn

Mapped to the Cambridge IGCSE XMA01-YMA01 syllabus (2026 onwards).

S2 1.1 — Understand and use the binomial distribution as a model.
S2 1.2 — Calculate $P(X = r)$ , $P(X \le c)$ , $P(X \ge c)$ using PMF, tables, or calculator.
S2 1.3 — Use $E(X) = np$ and $\text{Var}(X) = np(1-p)$ .
S2 1.4 — Identify and justify the four binomial conditions in context.
S2 1.5 — Solve inverse problems: find $c$ given a probability constraint.

Setup — when binomial is appropriate

Four conditions: fixed $n$ , two outcomes, independence, constant $p$ .

Definition. $X \sim B(n, p)$ is the count of successes in $n$ independent Bernoulli trials, each with success probability $p$ . The four conditions are:

Fixed number of trials $n$ . Not 'until something happens' — that would be geometric.
Two outcomes per trial. 'Success' (probability $p$ ) and 'failure' (probability $1 - p$ ).
Independence. The outcome of one trial does not affect another.
Constant $p$ . The success probability is the same for every trial.

Examples that fit.

Number of heads in $20$ tosses of a fair coin: $B(20, 0.5)$ .
Number of defective items in a batch of $50$ produced by an automated line with $2\%$ defect rate: $B(50, 0.02)$ .
Number of $6$ s rolled in $10$ throws of a fair die: $B(10, 1/6)$ .

Examples that DO NOT fit.

Number of cards drawn until you get an Ace (variable $n$ — geometric).
Number of red cards in $5$ draws WITHOUT replacement from a deck ( $p$ changes — hypergeometric).
Number of accidents per week (no fixed $n$ — Poisson).

Modelling caution. Real-world data often violates the conditions slightly. A binomial model can still be a useful approximation, but an examiner will reward critical comment.

Four conditions: fixed $n$ , two outcomes, independence, constant $p$ .
Notation: $X \sim B(n, p)$ .
Variable $n$ ? Use geometric. No replacement? Hypergeometric (not in S2).

PMF, cumulative, and tail probabilities

PMF gives $P(X = r)$ ; cumulative tables give $P(X \le c)$ ; tail uses complement.

PMF.

$P(X = r) = \binom{n}{r} p^{r} (1-p)^{n-r}, \quad r = 0, 1, \ldots, n.$

The $\binom{n}{r}$ counts the number of orderings; $p^{r}(1-p)^{n-r}$ is the probability of any one ordering with $r$ successes.

Cumulative.

$P(X \le c) = \sum_{r=0}^{c} \binom{n}{r} p^{r}(1-p)^{n-r}.$

The Edexcel booklet tabulates this for selected $n$ ( $5, 6, 7, 8, 9, 10, 12, 15, 20, 25, 30, 40, 50$ ) and $p \in \{0.05, 0.1, \ldots, 0.5\}$ . For other $(n, p)$ pairs, use calculator BinomialCD.

Upper tail (the key complement).

$P(X \ge c) = 1 - P(X \le c - 1).$

The $-1$ is mandatory: $P(X \ge 10) = 1 - P(X \le 9)$ , NOT $1 - P(X \le 10)$ .

Interval.

$P(a \le X \le b) = P(X \le b) - P(X \le a - 1).$

$P(X = r)$ from cumulative.

$P(X = r) = P(X \le r) - P(X \le r - 1).$

This is useful when tables are available but BinomialPD is not. With a calculator, use BinomialPD directly.

PMF: $\binom{n}{r} p^{r}(1-p)^{n-r}$ .
$P(X \le c)$ : cumulative table or BinomialCD.
$P(X \ge c) = 1 - P(X \le c - 1)$ .
$P(a \le X \le b) = P(X \le b) - P(X \le a - 1)$ .

Mean, variance, and standard deviation

$E(X) = np$ , $\text{Var}(X) = np(1-p)$ . SD is $\sqrt{npq}$ .

Formulae (in the booklet).

$E(X) = np, \quad \text{Var}(X) = np(1-p) = npq.$

Standard deviation. $\sigma = \sqrt{npq}$ . NOT in the booklet directly — derive from variance.

Worked example. $X \sim B(50, 0.04)$ .

$E(X) = 50 \times 0.04 = 2$ .
$\text{Var}(X) = 50 \times 0.04 \times 0.96 = 1.92$ .
$\sigma = \sqrt{1.92} \approx 1.386$ .

Why $np$ and $npq$ ?

A binomial $X = X_1 + X_2 + \ldots + X_n$ where each $X_i$ is a Bernoulli ( $0/1$ ) with mean $p$ and variance $p(1-p)$ .

$E(X) = \sum E(X_i) = np$ .
$\text{Var}(X) = \sum \text{Var}(X_i) = np(1-p)$ (because of independence — variances add).

Variance is maximised at $p = 0.5$ . The function $p(1-p)$ peaks at $p = 0.5$ with value $0.25$ . So $B(n, 0.5)$ is the 'most spread out' binomial for a given $n$ .

$E(X) = np$ — exam-ready in booklet.
$\text{Var}(X) = np(1-p)$ — exam-ready in booklet.
$\sigma = \sqrt{np(1-p)}$ — derive.
Variance maximised at $p = 0.5$ .

Inverse cumulative problems

Find smallest $c$ such that $P(X \le c) \ge p$ by tabulating adjacent values.

Setup. Find the smallest integer $c$ such that $P(X \le c) \ge 0.95$ (say). Equivalently, find the smallest $c$ such that $P(X \ge c + 1) < 0.05$ .

Method.

Tabulate $P(X \le 0), P(X \le 1), \ldots$ until you cross the threshold.
Identify the first $c$ where $P(X \le c) \ge 0.95$ .
Verify the previous value: $P(X \le c - 1) < 0.95$ .

Worked example. $X \sim B(30, 0.25)$ , find smallest $c$ with $P(X \ge c) < 0.05$ .

Equivalent: smallest $c$ with $P(X \le c - 1) > 0.95$ .
$P(X \le 11) = 0.9493$ — too low.
$P(X \le 12) = 0.9784$ — crosses.
So $c - 1 = 12 \Rightarrow c = 13$ .
Check. $P(X \ge 13) = 1 - 0.9784 = 0.0216 < 0.05$ . $P(X \ge 12) = 1 - 0.9493 = 0.0507 > 0.05$ . Confirmed.

Why this matters. This is exactly how a critical region is set up for a one-tail binomial hypothesis test (hypothesis testing topic).

Tabulate adjacent cumulative values.
Identify which $c$ crosses the threshold.
Verify both sides of the boundary.
Foundation of binomial hypothesis tests.

Modelling judgements

Check each of the four conditions in context. Independence and constant $p$ are usually the weak ones.

Standard exam prompt. 'State the conditions for the binomial model and discuss whether they hold in this context.'

Approach.

List the four conditions. Fixed $n$ , two outcomes, independence, constant $p$ .
For each, check against the context — quote details from the question.
Identify the WEAK ones — usually independence (e.g. people travelling together) or constant $p$ (e.g. variability across individuals).
Verdict. Is the model 'reasonable', 'unreasonable', or 'a first approximation'?

Example. $30$ students, $X$ = number attending a lecture.

Fixed $n = 30$ — holds.
Two outcomes (attend / not) — holds.
Independence — DOUBTFUL: friends travel together, illness can be contagious.
Constant $p$ — DOUBTFUL: students differ in engagement.

Verdict: 'A reasonable first approximation, but the independence and constant- $p$ assumptions are unlikely to hold exactly.'

What examiners reward.

Specific reasoning referencing the scenario.
A clear 'reasonable / unreasonable / approximate' verdict.

What examiners deduct for.

Generic statements ('might not be independent').
Listing without checking ('all four conditions hold' without evidence).

Always link each condition to context.
Independence and constant $p$ are usually the weak ones.
Use specific reasons — not generic.
End with a verdict.

How it’s examined

Binomial appears on every WST02/01 paper — typically a $4$ - $6$ mark short question on exact / cumulative probability, plus a longer modelling-judgement question ( $4$ - $5$ marks) and use in the binomial hypothesis test (later topic). The most common error is the off-by-one slip $P(X \ge c) = 1 - P(X \le c)$ — examiners flag this as the single most common WST02/01 mistake. Inverse-cumulative problems (find smallest $c$ such that $P(X \le c) \ge 0.95$ ) are the gateway to critical-region work in hypothesis testing. Calculator use is allowed throughout; BinomialPD and BinomialCD on a Casio FX-991 give exact values quickly.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WST02/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

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Step-by-step worked examples — Binomial Distributions

Step-by-step solutions to past-paper-style questions on binomial distributions, written exactly the way a tutor would explain them at the board.

1Single-point probability $P(X = r)$ (3 marks, S2)
Core• Adapted from WST02/01 January 2024 Q1• binomial, PMF, calculator
Question
A biased coin lands heads with probability $0.3$ . The coin is tossed $12$ times. Find $P(X = 4)$ , where $X$ is the number of heads.
Step-by-step solution
1. Step 1
  Identify the distribution. Independent tosses, fixed $n = 12$ , constant $p = 0.3$ , count of heads $\Rightarrow X \sim B(12, 0.3)$ .
2. Step 2
  Apply the PMF:
  $P(X = 4) = \binom{12}{4} (0.3)^{4} (0.7)^{8}$
3. Step 3
  $\binom{12}{4} = 495$ , $(0.3)^{4} = 0.0081$ , $(0.7)^{8} \approx 0.0576$ .
  $P(X = 4) \approx 495 \times 0.0081 \times 0.0576 \approx 0.2311$
Answer
$P(X = 4) \approx 0.2311$ ( $4$ dp).
Examiner tip
M1 for $\binom{12}{4} p^{4} q^{8}$ structure. A1 for $0.2311$ (accept $0.231$ ). Many calculators have BinomialPD $(r, n, p)$ which gives the answer directly — still write the formula to secure the M-mark.
2Cumulative probability $P(X \le c)$ (4 marks, S2)
Core• Adapted from WST02/01 June 2024 Q2• binomial, cumulative, tables
Question
$X \sim B(20, 0.4)$ . Find (a) $P(X \le 7)$ , (b) $P(X \ge 10)$ .
Step-by-step solution
1. Step 1
  (a) Read directly from the Edexcel cumulative binomial table (or calculator) for $n = 20$ , $p = 0.4$ , $r = 7$ .
  $P(X \le 7) = 0.4159$
2. Step 2
  (b) Convert $P(X \ge 10)$ using the complement of $P(X \le 9)$ .
  $P(X \ge 10) = 1 - P(X \le 9)$
3. Step 3
  From the table: $P(X \le 9) = 0.7553$ . Therefore
  $P(X \ge 10) = 1 - 0.7553 = 0.2447$
Answer
(a) $P(X \le 7) = 0.4159$ . (b) $P(X \ge 10) = 0.2447$ .
Examiner tip
(a) B1 for $0.4159$ . (b) M1 for $1 - P(X \le 9)$ , A1 for $0.2447$ . The single most common error: writing $1 - P(X \le 10)$ instead of $1 - P(X \le 9)$ for $P(X \ge 10)$ .
3Mean and variance, with $P(X \ge 5)$ (5 marks, S2)
Extended• Adapted from WST02/01 January 2024 Q3• binomial, mean, variance, context
Question
A factory produces components, $4\%$ of which are defective. A random sample of $50$ components is inspected. (a) State the distribution of $X$ , the number of defectives. (b) Find $E(X)$ and $\text{Var}(X)$ . (c) Find $P(X \ge 5)$ .
Step-by-step solution
1. Step 1
  (a) Independent components, fixed $n = 50$ , constant $p = 0.04$ .
  $X \sim B(50, 0.04)$
2. Step 2
  (b) Standard formulae from the booklet.
  $E(X) = np = 50 \times 0.04 = 2$
3. Step 3
  Variance:
  $\text{Var}(X) = np(1-p) = 50 \times 0.04 \times 0.96 = 1.92$
4. Step 4
  (c) Use the complement.
  $P(X \ge 5) = 1 - P(X \le 4)$
5. Step 5
  From calculator (table only goes to $p = 0.5$ in some editions but $p = 0.04$ requires calculator):
  $P(X \le 4) \approx 0.9510$
6. Step 6
  Evaluate:
  $P(X \ge 5) \approx 1 - 0.9510 = 0.0490$
Answer
(a) $X \sim B(50, 0.04)$ . (b) $E(X) = 2$ , $\text{Var}(X) = 1.92$ . (c) $P(X \ge 5) \approx 0.0490$ .
Examiner tip
(a) B1 for the distribution and parameters. (b) B1 each for mean and variance — formula booklet supplies $np$ and $np(1-p)$ . (c) M1 for $1 - P(X \le 4)$ , A1 for $0.0490$ (accept $0.049$ ). Examiners deduct for $1 - P(X \le 5)$ — a tail-direction error.
4Inverse cumulative — find smallest $c$ (5 marks, S2)
Extended• Adapted from WST02/01 June 2024 Q5• binomial, inverse, critical region
Question
$X \sim B(30, 0.25)$ . Find the smallest value of $c$ such that $P(X \ge c) < 0.05$ .
Step-by-step solution
1. Step 1
  Rewrite using the complement:
  $P(X \ge c) < 0.05 \;\Leftrightarrow\; P(X \le c - 1) > 0.95$
2. Step 2
  Tabulate (calculator or Edexcel tables) for $B(30, 0.25)$ :
3. Step 3
  $P(X \le 11) \approx 0.9493$ — just under $0.95$ . $P(X \le 12) \approx 0.9784$ — exceeds $0.95$ .
4. Step 4
  So we need $c - 1 = 12$ , i.e. $c = 13$ .
5. Step 5
  Check: $P(X \ge 13) = 1 - P(X \le 12) \approx 1 - 0.9784 = 0.0216 < 0.05$ . And $P(X \ge 12) = 1 - 0.9493 = 0.0507 > 0.05$ . Confirmed.
Answer
Smallest $c = 13$ .
Examiner tip
M1 for rewriting in cumulative form. M1 for tabulating both adjacent values. A1 for $c = 13$ . B1 for the explicit check that $c = 12$ fails. The 'just inside' boundary is the heart of critical-region work in hypothesis testing.
5Modelling validity — when is binomial appropriate? (5 marks, S2)
Extended• Adapted from WST02/01 January 2024 Q7• binomial, modelling, context
Question
In a class of $30$ students, $X$ is the number of students who attend a lecture on a particular day. Comment on whether $X \sim B(30, p)$ is a reasonable model. State THREE assumptions required for the binomial distribution and discuss each in this context.
Step-by-step solution
1. Step 1
  Assumption 1 — fixed number of trials. We have $n = 30$ students — fixed, so this holds.
2. Step 2
  Assumption 2 — two outcomes per trial (attends / does not attend). This holds.
3. Step 3
  Assumption 3 — independence of trials. Doubtful: friends may travel together; a contagious illness may make several students absent simultaneously; group projects can synchronise attendance. So independence is questionable.
4. Step 4
  Assumption 4 — constant probability $p$ . Also doubtful: keen students attend almost every lecture, less engaged students rarely do. So $p$ varies across students.
5. Step 5
  Verdict: the binomial model is a reasonable first approximation, but the independence and constant- $p$ assumptions are unlikely to hold exactly. Confidence intervals derived from $B(30, p)$ would underestimate true variability.
Answer
Three required assumptions: (i) fixed $n$ , (ii) independent trials, (iii) constant $p$ . In this context, fixed $n$ holds; independence is doubtful (shared travel / illness / group work); constant $p$ is doubtful (students differ in engagement). The model is a rough approximation only.
Examiner tip
B1 for each correctly stated assumption ( $3$ marks). B1 each for two reasoned contextual comments ( $2$ marks). Generic statements without context (e.g. 'might not be independent') score zero. Examiners want specific reasons referencing the scenario.

Key Formulae — Binomial Distributions

The formulae you need to memorise for binomial distributions on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Binomial PMF
$P(X = r) = \binom{n}{r} p^{r} (1-p)^{n-r}, \quad r = 0, 1, \ldots, n$
$n$
number of trials (positive integer)
$p$
probability of success on a single trial
$r$
observed number of successes
When to use
To find the exact probability of $r$ successes. IS in the IAL formula booklet — but using BinomialPD on a calculator is faster.
Example
$B(12, 0.3)$ , $P(X = 4) = \binom{12}{4} (0.3)^{4} (0.7)^{8} \approx 0.2311$ .
Mean and variance
$E(X) = np, \quad \text{Var}(X) = np(1-p) = npq$
$q$
$1 - p$ (probability of failure)
When to use
Whenever the question asks for mean / variance / SD of a binomial. IS in the IAL formula booklet.
Example
$B(50, 0.04)$ : $E(X) = 2$ , $\text{Var}(X) = 1.92$ , $\sigma = \sqrt{1.92} \approx 1.386$ .
Cumulative binomial
$P(X \le c) = \sum_{r=0}^{c} \binom{n}{r} p^{r} (1-p)^{n-r}$
$c$
upper limit of summation
When to use
Read from the Edexcel cumulative binomial tables (in the booklet for selected $n$ and $p$ ) or via calculator BinomialCD.
Example
$B(20, 0.4)$ : $P(X \le 7) = 0.4159$ from tables.
Upper-tail (complement)
$P(X \ge c) = 1 - P(X \le c - 1)$
$c$
smallest value in the tail
When to use
Whenever you want $P(X \ge c)$ from cumulative tables. Note the $c - 1$ — easy to forget.
Example
$P(X \ge 10) = 1 - P(X \le 9)$ .

Key Definitions and Keywords — Binomial Distributions

Definitions to memorise and the exact keywords mark schemes credit for binomial distributions answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Binomial distribution
Examiner keyword
Distribution of the number of successes $X$ in $n$ independent Bernoulli trials, each with success probability $p$ . Notation $X \sim B(n, p)$ .
Bernoulli trial
Examiner keyword
A single experiment with exactly two outcomes (success / failure) and constant success probability $p$ . The binomial is a sum of $n$ independent Bernoulli trials.
Binomial conditions
Examiner keyword
Four required assumptions: (i) fixed number of trials $n$ ; (ii) each trial has two outcomes; (iii) trials are independent; (iv) probability of success $p$ is constant across trials.
Cumulative probability
$P(X \le c)$ — probability that $X$ is at most $c$ . Read directly from Edexcel tables (where tabulated) or calculator.
Probability mass function (PMF)
Function giving $P(X = r)$ for each integer value $r$ in the support. For binomial: $\binom{n}{r} p^{r}(1-p)^{n-r}$ .

Common Mistakes and Misconceptions — Binomial Distributions

The traps other students keep falling into on binomial distributions questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Writing $P(X \ge c) = 1 - P(X \le c)$ instead of $1 - P(X \le c - 1)$
WST02/01 examiner reports — annual recurring error
Why it happens
Forgetting that $X = c$ is included in $P(X \ge c)$ but excluded from $P(X \le c - 1)$ .
How to avoid it
Always rewrite tail probabilities explicitly: $P(X \ge 10) = P(X = 10) + P(X = 11) + \ldots = 1 - P(X \le 9)$ .
✕Stating binomial conditions in generic terms without contextual reasoning
WST02/01 examiner reports
Why it happens
Students memorise 'independent and constant $p$ ' but don't apply it.
How to avoid it
Always link each assumption to the specific scenario: 'Trials are independent — but in this case students travel together, so independence may fail'.
✕Confusing variance $npq$ with standard deviation $\sqrt{npq}$
Why it happens
Both formulae are similar; students forget to take the square root for SD.
How to avoid it
Write 'Var $= np(1-p)$ ' first, then 'SD $= \sqrt{\text{Var}}$ ' as a second line.
✕Rounding $p$ , $\binom{n}{r}$ or intermediate products too early
Why it happens
Calculator displays are limited.
How to avoid it
Keep at least $5$ significant figures in intermediate working; round only the final answer to $4$ sf (or as the question asks).
✕Reading $P(X = 4)$ as $P(X \le 4)$ from tables
Why it happens
Edexcel tables list cumulative values; students miss the column heading.
How to avoid it
Use the PMF or BinomialPD for exact $P(X = r)$ ; use the cumulative table (or BinomialCD) for $P(X \le r)$ . Single point: $P(X = r) = P(X \le r) - P(X \le r - 1)$ .

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Binomial Distributions — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Binomial Distributions

Short Study Notes in the form of Slides

Detailed Notes

What you’ll learn

How it’s examined

1Single-point probability P(X=r)P(X = r)P(X=r) (3 marks, S2)

2Cumulative probability P(X≤c)P(X \le c)P(X≤c) (4 marks, S2)

3Mean and variance, with P(X≥5)P(X \ge 5)P(X≥5) (5 marks, S2)

4Inverse cumulative — find smallest ccc (5 marks, S2)

5Modelling validity — when is binomial appropriate? (5 marks, S2)

Binomial PMF

Mean and variance

Cumulative binomial

Upper-tail (complement)

Binomial distribution

Bernoulli trial

Binomial conditions

Cumulative probability

Probability mass function (PMF)

✕Writing P(X≥c)=1−P(X≤c)P(X \ge c) = 1 - P(X \le c)P(X≥c)=1−P(X≤c) instead of 1−P(X≤c−1)1 - P(X \le c - 1)1−P(X≤c−1)

✕Stating binomial conditions in generic terms without contextual reasoning

✕Confusing variance npqnpqnpq with standard deviation npq\sqrt{npq}npq​

✕Rounding ppp, (nr)\binom{n}{r}(rn​) or intermediate products too early

✕Reading P(X=4)P(X = 4)P(X=4) as P(X≤4)P(X \le 4)P(X≤4) from tables

Practice questions

Past paper style quiz

Binomial Distributions — frequently asked questions

1Single-point probability $P(X = r)$ (3 marks, S2)

2Cumulative probability $P(X \le c)$ (4 marks, S2)

3Mean and variance, with $P(X \ge 5)$ (5 marks, S2)

4Inverse cumulative — find smallest $c$ (5 marks, S2)

✕Writing $P(X \ge c) = 1 - P(X \le c)$ instead of $1 - P(X \le c - 1)$

✕Confusing variance $npq$ with standard deviation $\sqrt{npq}$

✕Rounding $p$ , $\binom{n}{r}$ or intermediate products too early

✕Reading $P(X = 4)$ as $P(X \le 4)$ from tables