IB Maths AI SLBinomial DistributionPaper 1 & 2GDC: Binomial PD / CD~6 min read
Calculating Binomial Probabilities
Once you know X ~ B(n, p), the GDC does the arithmetic. The real skill is reading the question — “at least”, “fewer than”, “more than” — and turning those words into the right range of whole numbers, then picking Binomial PD or CD.
📘 What you need to know
One exact value, P(X = x): use the GDC’s Binomial PD function (also called BPD or Binomial Pdf) — enter x, n, p.
A range, P(X ≤ x) or P(a ≤ X ≤ b): use the Binomial CD function (BCD or Binomial Cdf) — enter the lower value, upper value, n, p.
X is a whole number: 0, 1, 2, …, n. There is no P(X = 4.5) — it is always 0.
Translate the words: “at least a” ⇒ X ≥ a · “at most b” ⇒ X ≤ b · “more than a” ⇒ X > a · “fewer than b” ⇒ X < b.
Strict to weak (binomial only): P(X < x) = P(X ≤ x−1) and P(X > x) = P(X ≥ x+1).
Complement: P(X ≥ a) = 1 − P(X ≤ a−1) — the fastest route for “at least”.
Single probabilities: P(X = x)
P(X = x) is the probability of exactlyx successes. On your GDC this is the Binomial PD function — you enter the value x, the number of trials n, and the success probability p.
Single binomial probability
P(X = x) = nCx × px × (1 − p)n−xthe GDC’s Binomial PD function does this for you — the formula is in the booklet
You are expected to use the GDC function, not the formula by hand. Always define X in words first so it is clear what one success means.
Cumulative probabilities: P(X ≤ x) and ranges
A cumulative probability adds up several values at once. The Binomial CD function finds P(a ≤ X ≤ b) directly: you enter a lower value a, an upper value b, then n and p.
Two special cases: for P(X ≤ b) set the lower value to 0 · for P(X ≥ a) set the upper value to n. The chart below shows what “cumulative” means — the total height of a block of bars.
For X ~ B(15, 0.5), the cumulative probability P(5 ≤ X ≤ 9) is just the five shaded bars added together — that is exactly what Binomial CD computes.
If your calculator only finds P(X ≤ x), these identities rebuild any other range:
Cumulative identities for a binomial
P(X < x) = P(X ≤ x−1) · P(X > x) = 1 − P(X ≤ x)
P(X ≥ x) = 1 − P(X ≤ x−1)P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a−1)
Turning words into a range of integers
Most marks are lost here, not on the GDC. The trick: decide the smallest and biggest whole number the question allows, then read off the range a ≤ X ≤ b.
Watch the endpoint: “at least 6” and “at most 6” include 6 · “more than 6” means 7 or above · “fewer than 6” means 5 or below. Because X is a whole number, every strict inequality becomes a weak one by shifting the endpoint by 1.
So “more than 8 but fewer than 15” becomes 9 ≤ X ≤ 14 — a clean range your GDC can take straight away.
🧭 Recipe — any binomial probability question
State the model: write X ~ B(n, p) and say in words what X counts.
Underline the wording: exactly · at least · at most · more than · fewer than · between.
Write the range: turn the words into X = x or a ≤ X ≤ b — shift any strict inequality by 1.
Pick the function: one exact value → Binomial PD · a range → Binomial CD (use the complement if “at least”).
Enter n, p and the range; write the answer in context, rounded to 3 significant figures.
Worked examples
WE 1
A single value — P(X = x)
A basketball player makes 75% of her free throws. In one game she takes 12 free throws. Let X be the number she makes. Find the probability that she makes exactly 9.
model: X ~ B(12, 0.75)“exactly 9” ⇒ one value, X = 9use Binomial PD: x = 9, n = 12, p = 0.75P(X = 9) = 12C9 × 0.759 × 0.253P(X = 9) ≈ 0.258“Exactly” always means a single value — Binomial PD, not CD. About a 26% chance she makes exactly 9.
WE 2
“At most” — P(X ≤ x)
A seed packet claims 80% of its seeds germinate. A gardener plants 15 seeds. Let X be the number that germinate. Find the probability that at most 11 germinate.
model: X ~ B(15, 0.8)“at most 11” ⇒ X ≤ 11 (11 is included)use Binomial CD: lower = 0, upper = 11, n = 15, p = 0.8P(X ≤ 11) = P(0 ≤ X ≤ 11)P(X ≤ 11) ≈ 0.352“At most” includes its endpoint, so 11 stays in. For a P(X ≤ x) range, enter 0 as the lower value.
WE 3
“At least” — the complement
A marketing team finds that 30% of its emails are opened. A campaign sends 20 emails. Let X be the number opened. Find the probability that at least 8 are opened.
model: X ~ B(20, 0.3)“at least 8” ⇒ X ≥ 8use the complement: P(X ≥ 8) = 1 − P(X ≤ 7)P(X ≤ 7) ≈ 0.7723 (Binomial CD)P(X ≥ 8) = 1 − 0.7723P(X ≥ 8) ≈ 0.228“At least 8” is 8 or more, so you cut off X ≤ 7 — note the 7, not 8. Subtracting from 1 is the quickest route.
WE 4
A strict “between” — shift the endpoints
On a production line 4% of light bulbs are defective. An inspector checks a box of 50. Let X be the number of defective bulbs. Find P(1 < X < 6).
model: X ~ B(50, 0.04)strict inequalities — shift each endpoint by 11 < X < 6 ⇒ 2 ≤ X ≤ 5use Binomial CD: lower = 2, upper = 5, n = 50, p = 0.04P(1 < X < 6) = P(2 ≤ X ≤ 5) ≈ 0.585because X is a whole number, “> 1” means “≥ 2” and “< 6” means “≤ 5”. Convert before you touch the GDC.
WE 5
Reading “more than” and “fewer than”
On a bus route, 85% of buses arrive on time. A commuter records 30 journeys. Let X be the number of on-time buses. Find the probability that (a) fewer than 25 are on time, (b) more than 26 are on time.
model: X ~ B(30, 0.85)(a) “fewer than 25” ⇒ X < 25 ⇒ X ≤ 24P(X ≤ 24) ≈ 0.289 (Binomial CD)(b) “more than 26” ⇒ X > 26 ⇒ X ≥ 27P(X ≥ 27) = 1 − P(X ≤ 26) ≈ 1 − 0.678(a) ≈ 0.289 · (b) ≈ 0.322“Fewer than 25” excludes 25 (X ≤ 24); “more than 26” excludes 26 (X ≥ 27). “Than” never includes its own number.
WE 6
Full question: model and three probabilities
A coffee shop finds that 40% of its customers use its loyalty app. The manager samples 25 customers. Let X be the number who use the app. (a) State the distribution. (b) Find P(X = 10). (c) Find the probability that at most 8 use the app. (d) Find the probability that more than 12 use the app.
(a) state the modelX ~ B(25, 0.4)(b) one value → Binomial PDP(X = 10) ≈ 0.161(c) “at most 8” ⇒ X ≤ 8 → Binomial CDP(X ≤ 8) ≈ 0.274(d) “more than 12” ⇒ X ≥ 13P(X > 12) = 1 − P(X ≤ 12) ≈ 0.154(a) X ~ B(25, 0.4) · (b) 0.161 · (c) 0.274 · (d) 0.154spot the function from the wording: “= 10” is PD; “at most” and “more than” are CD ranges. For (d), cutting off X ≤ 12 leaves X ≥ 13.
💡 Top tips
Write the inequality down — even in a worded question. You can earn a method mark for P(X ≥ 27) even if you mistype on the GDC.
“At least” / “at most” keep the number; “more than” / “fewer than” drop it. This single rule fixes most lost marks.
Binomial PD for one value, Binomial CD for a range — match the function to the wording before you start typing.
Use the complement for “at least”: P(X ≥ a) = 1 − P(X ≤ a−1) is faster and less error-prone.
Keep full accuracy — round only the final answer, normally to 3 significant figures.
⚠ Common mistakes
Off-by-one errors: “more than 5” is X ≥ 6, not X ≥ 5 — the most common slip in the whole topic.
Wrong complement: P(X ≥ a) = 1 − P(X ≤ a−1), not 1 − P(X ≤ a).
P(X < x) treated as P(X ≤ x): a strict “<” must drop to x−1 first.
Mixing up PD and CD: Binomial PD gives one bar; Binomial CD gives a running total of bars.
Rounding too early: a value rounded mid-calculation can shift the final answer in the last digit.
The biggest source of lost marks here is the off-by-one in “more than” and “at least”. Rewrite the request as a clean range a ≤ X ≤ b before touching the GDC — once the range is right, the calculator won’t let you down.
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