IB Maths AA SL Topic 4 — Probability Distributions Paper 2 (GDC) ~10 min read

Calculating Binomial Probabilities

Once you’ve written X ∼ B(n, p), the maths is mostly done — your calculator does the heavy lifting. The real skill on this topic is reading the question carefully and turning words like “at least” or “fewer than” into the right numbers to type in. Get that translation right and these become some of the easiest marks on Paper 2.

📘 What you need to know

Use Binomial Pdf on your GDC for one specific value: P(X = x).
Use Binomial Cdf for a range of values: P(a ≤ X ≤ b).
Convert strict inequalities (<, >) into ≤ and ≥ before typing — e.g. P(X < 5) becomes P(X ≤ 4).
Use the complement for “at least” or “more than”: P(X ≥ k) = 1 − P(X ≤ k − 1).
“At least one” is almost always 1 − P(X = 0).
Always write the inequality on paper before running the calculator — that’s where method marks live.

Pdf or Cdf? Just two options

Whatever the question, you’ll only ever pick between two functions on your calculator. The rule is simple: if you want the chance of one specific value, use the first one. If you want the chance over a range of values, use the second.

Picture the binomial distribution as a row of vertical bars on a graph — each bar shows the probability of one value of X:

Binomial Pdf gives you the height of one bar — that’s P(X = x).
Binomial Cdf gives you the total height of a chunk of bars from a to b added together.

If a question says “exactly”, “equal to”, or names a single number — use Pdf. If it talks about “between”, “at most”, “fewer than”, or “at least” — use Cdf (often combined with the complement trick we’ll get to in a minute).

What to type into your calculator

Each function needs a few pieces of info from you. The buttons live in the Distribution menu on your GDC — look for “Binomial Pdf” and “Binomial Cdf”. To show how it works, here’s exactly what you’d put in for the example we’ll see in WE 1 and WE 3, where X ∼ B(40, 0.35).

Binomial Pdf single value

x value10

n (trials)40

p0.35

P(X = 10) ≈ 0.0571

Binomial Cdf range of values

Lower (a)9

Upper (b)14

n (trials)40

p0.35

P(9 ≤ X ≤ 14) ≈ 0.542

📍

If your calculator only does P(X ≤ b)

Some older calculators can only handle the “from 0 up to b” version. Don’t worry — you can still find any range using subtraction: P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1). Just two calculator runs.

The “< and >” problem

Here’s something that catches students out every year. Your GDC’s Cdf function only takes ≤ and ≥ — it doesn’t understand strict inequalities. So when you see X < 5 or X > 8 in a question, you have to rewrite it first.

The good news is that because X can only take whole-number values (0, 1, 2, 3, …), this conversion is super simple. Just shift by 1.

If the question gives…	Type into the GDC…	Why
P(X < k)	P(X ≤ k − 1)	k isn’t included
P(X > k)	1 − P(X ≤ k)	flip with the complement
P(X ≥ k)	1 − P(X ≤ k − 1)	complement of “less than k“
P(a < X ≤ b)	P(a + 1 ≤ X ≤ b)	a excluded → bump up
P(a ≤ X < b)	P(a ≤ X ≤ b − 1)	b excluded → bump down
P(a < X < b)	P(a + 1 ≤ X ≤ b − 1)	bump both endpoints

🧠

An easy way to remember it

If the symbol has a line under it (≤ or ≥), the number is included. If it doesn’t (< or >), the number is excluded — and you shift by 1 to push it out of your range. That’s literally the whole rule.

If you ever get confused, list out the actual values. P(2 < X < 6) means {3, 4, 5}. So in the GDC: lower = 3, upper = 5. Listing kills any doubt.

The complement trick

This is the move that turns long, painful questions into short ones. Whenever you see “at least“, “more than“, or especially “at least one“, the complement is almost always faster than going the long way round.

The idea: instead of adding up all the probabilities you want, find the probability of everything you don’t want, then subtract from 1.

Complement formulas

P(X ≥ k) = 1 − P(X ≤ k − 1)

P(X > k) = 1 − P(X ≤ k)

“At least one” — the classic exam question

You’ll see this exact phrasing on almost every Paper 2: “What is the probability of at least one success?” The opposite of “at least one” is “none” — and that’s just one Pdf calculation. So:

“At least one” shortcut

P(X ≥ 1) = 1 − P(X = 0)

🤔 Why this saves so much time

Imagine X ∼ B(40, 0.35) and someone asks for P(X ≥ 18). Without the complement, you’d have to add P(18) + P(19) + P(20) + … all the way up to P(40). That’s 23 separate numbers. With the complement, it’s just 1 − P(X ≤ 17) — one calculator run.

The 3 steps to follow every time

If you do these three steps in this order on every binomial probability question, you’ll almost never go wrong:

Write down the inequality you want in its exact form — e.g. “P(X ≥ 8)”.
Convert it into something your GDC can take (≤ or ≥, possibly with a complement).
Run the GDC and write the answer to 3 significant figures (unless told otherwise).

Don’t skip step 1. Even if it feels obvious, writing the inequality on paper protects you. If you mistype something on the calculator, the examiner can still see your method and award method marks.

Worked examples

WE 1

Find a single-value probability

The random variable X ∼ B(40, 0.35). Find P(X = 10), giving your answer to 3 s.f.

“= 10” means a single value, so use Binomial Pdf. Pull out: n = 40, p = 0.35, x = 10 Run Binomial Pdf: P(X = 10) = 0.057056… P(X = 10) = 0.0571 writing n, p, and x on the page first earns method marks even if your final number is off!

WE 2

“At most” — a cumulative probability

For X ∼ B(40, 0.35), find P(X ≤ 10).

“At most 10” includes 10 → range from 0 to 10. Use Binomial Cdf. Rewrite as range: P(X ≤ 10) = P(0 ≤ X ≤ 10) Run Binomial Cdf with lower = 0, upper = 10 P(X ≤ 10) = 0.121491… P(X ≤ 10) = 0.121 “at most” includes the value — that ≤ is locked in!

WE 3

Convert strict inequalities first

For X ∼ B(40, 0.35), find P(8 < X < 15).

Both 8 and 15 are excluded — bump lower up by 1, upper down by 1. Convert: P(8 < X < 15) = P(9 ≤ X ≤ 14) Run Binomial Cdf with lower = 9, upper = 14 P(9 ≤ X ≤ 14) = 0.541827… P(8 < X < 15) = 0.542 always rewrite the inequality on paper FIRST — never just do it in your head!

WE 4

“At least” — use the complement

For X ∼ B(40, 0.35), find P(X ≥ 18).

Adding from 18 to 40 = 23 numbers. Use the complement instead. Flip with complement: P(X ≥ 18) = 1 − P(X ≤ 17) Run Binomial Cdf with lower = 0, upper = 17 P(X ≤ 17) = 0.929945… P(X ≥ 18) = 1 − 0.929945… = 0.070054… P(X ≥ 18) = 0.0701 complement turns 23 calculations into just 1 — always look for it!

WE 5

A real exam-style question

A multiple-choice quiz has 12 questions, each with 4 options (only one correct). Sara guesses every answer. Let X be the number she gets right.

(a) State the distribution of X.
(b) Find the probability she gets exactly 5 right.
(c) Find the probability she gets at least one right.
(d) Find P(2 ≤ X < 7).

Each guess is independent with p = 1/4 of being right → binomial.part (a) — set it up 12 trials, p = 0.25 X ∼ B(12, 0.25)part (b) — exactly 5 Single value → Binomial Pdf, x = 5 P(X = 5) = 0.10322… P(X = 5) = 0.103part (c) — at least one Complement: P(X ≥ 1) = 1 − P(X = 0) Pdf with x = 0: P(X = 0) = 0.03168… P(X ≥ 1) = 1 − 0.03168… = 0.96832… P(X ≥ 1) = 0.968part (d) — mixed inequality 7 is excluded: P(2 ≤ X < 7) = P(2 ≤ X ≤ 6) Cdf with lower = 2, upper = 6 = 0.84200… P(2 ≤ X < 7) = 0.842 just guessing → almost certain to get at least one right (~97%) but rare to get 5+. that’s the binomial in action!

💡 Top tips

Always write the inequality on paper first. e.g. “P(X ≥ 8) = 1 − P(X ≤ 7)”. This is where method marks come from.
Pdf for one value, Cdf for a range. If you remember nothing else, remember this.
Convert < and > into ≤ and ≥ before going to the calculator. Just shift by 1.
If you see “at least”, “more than”, or “at least one” — try the complement first. Almost always faster.
“At least one” = 1 − P(X = 0). Memorise this exact line.
Round only at the very end — usually 3 s.f. unless the question says otherwise.
Write down n and p next to your answer. It’s a free method mark.
If stuck, list the actual values (e.g. “between 5 and 9 inclusive” = {5, 6, 7, 8, 9}). It clears up any inequality confusion.

⚠ Common mistakes

Confusing Pdf and Cdf. If you put P(X ≤ 8) into Pdf, you’ll only get P(X = 8) — totally different value.
Forgetting to convert < or > before typing. P(X < 5) is NOT the same as P(X ≤ 5) — it’s P(X ≤ 4).
Skipping the complement on “at least” questions. Adding 20+ probabilities by hand is slow and easy to mess up.
Off-by-one errors. P(X > 5) = 1 − P(X ≤ 5), not 1 − P(X ≤ 4). Read the symbol carefully.
Including the wrong endpoint in P(a < X < b). If both are strict, both must be shifted.
Rounding too early. Carry the full decimal through your working — only round the final answer.
Typing p as a percentage. 35% means p = 0.35, not 35.
Mixing up n and x. n is the total number of trials; x is the success count you’re testing.

🎉 You can now handle every binomial probability question that Paper 2 throws at you. Next up in Topic 4: the Normal Distribution — basically the same idea but for continuous data. The good news is that once you’ve got the GDC routine down here, the normal distribution feels almost identical.

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