IB Maths AI HLBinomial DistributionPaper 1 & 2~7 min read
Calculating Binomial Probabilities
Once a situation is modelled as X ∼ B(n, p), the actual probabilities come from your GDC — not by hand. Use the binomial PD for a single value P(X = x) and the binomial CD for a range P(a ≤ X ≤ b). The only real skill is turning the inequality in the question (which may be strict, one-sided, or open) into a clean integer range your calculator can take.
📘 What you need to know
X is integer-valued: P(non-integer) = P(negative) = 0. All values are 0, 1, 2, …, n.
Single value: use binomial PD (a.k.a. BPD / Binomial Pdf) for P(X = x).
Range: use binomial CD (a.k.a. BCD / Binomial Cdf) for P(a ≤ X ≤ b).
Inputs: PD needs x, n, p; CD needs lower, upper, n, p.
If your GDC only does P(X ≤ b): combine with 1 − … and subtraction (identities below).
Write the inequality down as well as the answer — it can earn method marks.
Single value: P(X = x)
For one exact value, use the binomial probability distribution function on your GDC. There’s a formula too, but the exam expects the calculator.
Probability of a single value
P(X = x) = nCxpx(1 − p)n−xuse the binomial PD on your GDC
🧭 Recipe — P(X = x) on the GDC
Identifyn and p from the distribution B(n, p).
Open the binomial PD function (BPD / Binomial Pdf).
Enter the x value, then n, then p.
Read off the probability and round (usually 3 sf).
WE 1
A single value
X ∼ B(40, 0.35). Find P(X = 10).
identify n and pn = 40, p = 0.35use binomial PD on GDCP(X = 10) = 0.057056…P(X = 10) ≈ 0.057 (3sf)enter x = 10, n = 40, p = 0.35 into the BPD function.
Cumulative: P(a ≤ X ≤ b)
For a range of values, use the binomial cumulative distribution function. Most GDCs let you enter a lower and upper limit directly.
Open-ended ranges: if there’s no lower limit use 0; if there’s no upper limit use n. So P(X ≤ b) = P(0 ≤ X ≤ b) and P(X ≥ a) = P(a ≤ X ≤ n).
🤔 Why does P(X < 5) become P(X ≤ 4)?
Because X only takes whole numbers, “less than 5” means the integers 0, 1, 2, 3, 4 — exactly the same set as “≤ 4”. This integer trick only works for discrete variables like the binomial; it would not hold for a continuous variable. The same logic gives P(X > 5) = P(X ≥ 6).
🧭 Recipe — pin down the integer range
Find the smallest integerX can take in the range (the lower limit a).
Find the largest integer in the range (the upper limit b).
Entera, b, n, p into the binomial CD function.
Write the rewritten inequality next to your answer for method marks.
WE 2
A “≤” cumulative probability
X ∼ B(40, 0.35). Find P(X ≤ 10).
rewrite with a lower limitP(X ≤ 10) = P(0 ≤ X ≤ 10)use binomial CD on GDCP(X ≤ 10) = 0.121491…P(X ≤ 10) ≈ 0.121 (3sf)lower = 0, upper = 10, n = 40, p = 0.35.
WE 3
A strict double inequality
X ∼ B(40, 0.35). Find P(8 < X < 15).
identify upper and lower integersP(8 < X < 15) = P(9 ≤ X ≤ 14)use binomial CD on GDCP(9 ≤ X ≤ 14) = 0.541827…P(8 < X < 15) ≈ 0.542 (3sf)strict both ends: bump the lower up by 1 and the upper down by 1.
Inequality identities
If your GDC only gives P(X ≤ x), every other probability can be built from it. These hold because X is a binomial (integer) variable.
You want
Use P(X ≤ …)
Example
P(X < x)
P(X ≤ x − 1)
P(X < 5) = P(X ≤ 4)
P(X > x)
1 − P(X ≤ x)
P(X > 5) = 1 − P(X ≤ 5)
P(X ≥ x)
1 − P(X ≤ x − 1)
P(X ≥ 5) = 1 − P(X ≤ 4)
P(a ≤ X ≤ b)
P(X ≤ b) − P(X ≤ a − 1)
P(5 ≤ X ≤ 9) = P(X ≤ 9) − P(X ≤ 4)
Turning any inequality into an integer range
List the integers actually included, then read off the smallest and largest.
WE 4
Using “greater than”
X ∼ B(40, 0.35). Find P(X > 15).
rewrite as 1 − cumulativeP(X > 15) = 1 − P(X ≤ 15)GDC: P(X ≤ 15)= 0.749659… → 1 − 0.749659…P(X > 15) ≈ 0.250 (3sf)“greater than 15” excludes 15, so subtract P(X ≤ 15) from 1.
WE 5
A mixed open/closed range
X ∼ B(40, 0.35). Find P(12 ≤ X < 18).
find the integer rangeP(12 ≤ X < 18) = P(12 ≤ X ≤ 17)use binomial CD on GDClower = 12, upper = 17, n = 40, p = 0.35P(12 ≤ X < 18) ≈ 0.700 (3sf)“≤” keeps 12; strict “< 18” drops 18, giving an upper of 17.
💡 Top tips
PD for one value, CD for a range — pick the right GDC function first.
List the integers in the range, then take the smallest and largest as a and b.
Strict < / > shift the limit by 1: P(X < x) = P(X ≤ x − 1), P(X > x) = P(X ≥ x + 1).
No lower/upper limit? use 0 or n.
Write the inequality down beside the final answer — method marks if you mistype.
Keep extra decimals in working and round only at the end (usually 3 sf).
⚠ Common mistakes
Confusing < with ≤ — forgetting to shift the limit by 1 for strict inequalities.
Using PD for a range (or CD for a single value) — wrong function entirely.
Off-by-one on P(X ≥ x) — it’s 1 − P(X ≤ x − 1), not 1 − P(X ≤ x).
Swapping lower and upper limits in the CD function.
Rounding too early, then losing accuracy in a follow-up calculation.
Not stating the rewritten inequality, so a mistyped GDC entry earns nothing.
That completes the Binomial Distribution unit! You can now check the four conditions, set up X ∼ B(n, p), find its mean and variance, and calculate exact and cumulative probabilities on your GDC. Next this same GDC-driven approach carries into the Poisson and normal distributions, where you’ll meet new shapes but the very same PD / CD workflow.
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