Assumptions and Limitations
A bag of $20$ marbles contains $8$ red. You draw $5$ marbles without replacement. Can you use the binomial distribution? A surveyor asks people in a queue whether they recycle, but the people are friends in a group. Can you use binomial here? These questions probe the four conditions the binomial requires — and what to do when one of them fails.
From memory: list every condition you can think of that must hold for $X \sim \text{Bin}(n, p)$ to be the right model. Aim for at least three. Don't worry about precise wording yet.
The binomial distribution is only valid when all four conditions hold. A useful mnemonic is BINS: Binary outcomes, Independent trials, fixed Number of trials, and Same probability of success on each trial. If even one of these fails, the binomial formula gives wrong probabilities.
The BINS checklist for a binomial scenario:
B — Binary: each trial has exactly two outcomes (success / failure).
I — Independent: the outcome of one trial does not affect the others.
N — Fixed $n$: the number of trials is decided in advance.
S — Same $p$: the probability of success is identical on every trial.
Key facts
- The four BINS conditions: Binary, Independent, fixed $N$, same $p$
- Why sampling without replacement breaks independence and changes $p$
- The large-population rule: if pop $> 10n$, binomial is a good approximation
Concepts
- Why all four conditions matter — failing any one invalidates the model
- How to identify which condition fails in a real-world scenario
- What alternative distributions apply (geometric, hypergeometric) when binomial fails
Skills
- Test a scenario against all four BINS conditions, ticking off each one
- Identify exactly which condition fails when a binomial isn't appropriate
- Decide when the large-population approximation makes binomial usable anyway
For every binomial question, run a quick BINS check before writing any formula:
- B — Is each trial binary? Define "success" and "failure" clearly.
- I — Are the trials independent? Sampling with replacement, or from a large population, usually qualifies.
- N — Is $n$ fixed in advance? "Until the first success" means $n$ is random — geometric, not binomial.
- S — Is $p$ constant on every trial? Watch for scenarios where conditions change between trials.
Worked through the hook: Drawing $5$ cards without replacement, count hearts.
- B ✓ — Each card is either a heart or not.
- I ✗ — The first card's outcome changes the deck composition for the second draw.
- N ✓ — $n = 5$ is fixed.
- S ✗ — Initial $p = 13/52 = 1/4$, but after drawing a heart $p = 12/51 \neq 1/4$.
- So without replacement, this is not binomial. With replacement, all four pass: $X \sim \text{Bin}(5, 0.25)$.
For every binomial question, run a quick BINS check before writing any formula:
Pause — copy the BINS checklist as four yes/no questions and work through the hook example to identify which letters pass into your book.
Quick check: A jar contains $10$ red and $10$ blue marbles. You draw $4$ marbles without replacement. Let $X$ be the number of red marbles drawn. Which BINS condition is the main reason this is not a true binomial?
We just saw the BINS checklist: B (binary outcomes), I (independent trials), N (fixed number of trials), S (same $p$). That raises a question: when one condition fails, what specific error does it introduce — and which alternative model becomes appropriate instead? This card answers it → I fails → hypergeometric model; N fails → geometric model; B fails → multinomial; S fails → no standard model.
Each failed condition corrupts the binomial in a specific, predictable way:
- B fails (more than two outcomes): you cannot define "success" cleanly. Solution: regroup outcomes into two categories first (e.g. "rolled a six" vs "didn't roll a six").
- I fails (trials affect each other): binomial overstates the variance. Use hypergeometric for finite-population sampling.
- N fails ($n$ random): binomial cannot describe a random number of trials. If you're counting trials until the first success, use the geometric distribution.
- S fails ($p$ varies): binomial assumes a single $p$. If $p$ differs across trials, you have a "Poisson binomial" — there is no clean formula and you usually must enumerate cases.
Each failed condition corrupts the binomial in a specific, predictable way:
Pause — copy the four failure modes with their consequences: I fails (dependence), N fails (variable length), B fails (multi-outcome), S fails (changing $p$) into your book.
Did you get this? True or false: a die is rolled until a six appears for the first time, and $X$ is the number of rolls required. Then $X$ follows a binomial distribution.
Worked examples · 3 in a row, reveal as you go
A free-throw shooter has a long-run success rate of $0.8$. She attempts $25$ free throws, and $X$ counts the successes. Justify whether $X \sim \text{Bin}(25, 0.8)$ is an appropriate model by checking each BINS condition.
A bag contains $6$ red and $4$ green balls. Three balls are drawn without replacement. Let $X$ be the number of red balls drawn. Explain why $X$ is not binomially distributed, and state which BINS condition(s) fail.
A city of $2{,}000{,}000$ adults has support rate $0.4$ for a policy. A pollster samples $n = 800$ adults without replacement and counts supporters. Is the binomial model appropriate? Justify.
Fill the gap: The four BINS conditions for a binomial model are: outcomes, trials, fixed , and same probability $p$ on every trial.
Misconceptions to fix · the 3 traps that cost marks
Did you get this? True or false: in a population of $50{,}000$ people, sampling $n = 100$ without replacement still gives an excellent binomial approximation because the population is far larger than the sample.
Activities · practice with the ideas
For each scenario, decide if a binomial model is appropriate and justify by listing each BINS condition: (i) flipping a fair coin $20$ times and counting heads; (ii) drawing $4$ cards from a $52$-card deck without replacement and counting aces.
Explain in one or two sentences why "the number of dice rolls until the first six" is not a binomial random variable. Which condition fails?
A pollster samples $200$ voters from a city of $1$ million without replacement. The true support rate for a candidate is $0.45$. Is binomial appropriate? Justify using the large-population rule.
A factory worker inspects $50$ items chosen from a batch of $80$, without replacement. The defective rate in the batch is $5\%$. Identify which BINS condition(s) fail and whether the large-population rule rescues the binomial.
An archer takes $10$ shots at a target. The probability of hitting the target increases by $0.02$ on each successive shot as she warms up (so $p_1 = 0.5, p_2 = 0.52, \dots, p_{10} = 0.68$). Is this a binomial situation? If not, which condition fails?
Odd one out: Three of these scenarios are valid binomial random variables. Which one is NOT?
Earlier you listed the conditions you thought the binomial required.
The complete list is BINS: Binary outcomes, Independent trials, fixed Number of trials, and Same probability of success on each trial. The most commonly missed conditions are I (independence) and S (constant $p$), which both fail in without-replacement sampling from small populations. Whenever you see "without replacement", run the pop/$n$ check.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. State the four conditions (BINS) that must hold for a random variable $X$ to be modelled by $\text{Bin}(n, p)$. (2 marks)
Q2. A bag contains $5$ red and $5$ blue marbles. Four marbles are drawn without replacement. Let $X$ be the number of red marbles drawn. Explain whether $X \sim \text{Bin}(4, 0.5)$, naming any condition(s) that fail. (3 marks)
Q3. A research team is studying flu transmission in classrooms of $25$ students. They model the number of infected students per classroom as $\text{Bin}(25, 0.1)$. Identify two reasons (i.e. two BINS conditions) that are likely violated in this setting, and explain how each one might fail. (3 marks)
Comprehensive answers (click to reveal)
Activity answers:
1. (i) All four BINS hold ($n=20$, $p=0.5$, independent flips, two outcomes) — binomial valid. (ii) Without replacement from a small deck: I and S fail (drawing one ace changes the deck and the probability for the next draw). Pop/n = 52/4 = 13 — borderline; binomial is approximate but not exact.
2. N fails. The number of rolls is random (depends on when the first six occurs), not fixed in advance. This is a geometric distribution, $\text{Geom}(1/6)$.
3. Pop/n = $10^6/200 = 5000 \gg 10$. Removing $200$ people from a million barely shifts $p$. I and S hold approximately, so $X \approx \text{Bin}(200, 0.45)$ is an excellent model.
4. Pop/n = $80/50 = 1.6 \ll 10$. Without replacement, I and S fail markedly. Binomial inappropriate — use hypergeometric for an exact answer.
5. S fails. $p$ varies across the $10$ shots ($p_1=0.5$ up to $p_{10}=0.68$). Not binomial. (This is a Poisson binomial, beyond Ext 1 scope, but the failure of S is the key observation.)
Q1 (2 marks): B – each trial has exactly two outcomes (success/failure); I – trials are independent; N – $n$ is fixed in advance; S – the success probability $p$ is the same on every trial [1 mark for at least 3, 2 marks for all 4 clearly stated].
Q2 (3 marks): Without replacement: I fails because each draw changes the bag composition [1]. S fails because $p$ changes from $0.5$ to either $4/9$ or $5/9$ after the first draw, etc. [1]. Pop/$n = 10/4 = 2.5 \ll 10$ so the large-population rescue does not apply — $X \not\sim \text{Bin}(4, 0.5)$ [1].
Q3 (3 marks): I fails: students in the same classroom interact, so once one student is infected the probability others become infected increases — trials are not independent [1.5]. S fails: students have different immunity levels (vaccination, prior infection, age), so $p$ is not constant across the class [1.5]. (Either I or S would also be acceptable; full marks for both clearly stated with reasoning.)
Five timed scenario questions where you have to identify whether the binomial applies and, if not, which condition fails. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering BINS-check questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.