Cumulative Probabilities
A single binomial term is rarely what a question actually wants — "at least", "no more than", "fewer than" all describe a whole range of outcomes, and cumulative probabilities are how you add them up in one step.
The cumulative distribution function
For a random variable $X$, the cumulative probability function gives the total probability of $X$ taking a value up to and including $x$:
For $X\sim B(n,p)$, tables of $P(X\le x)$ exist for common values of $n$ and $p$, and every scientific or graphical calculator has a binomial cumulative distribution function built in — so in practice you almost never sum individual terms by hand once $x$ is more than 1 or 2.
A short extract
Here is $P(X\le x)$ for $X\sim B(6,0.4)$, computed term by term:
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| P(X≤x) | 0.0467 | 0.2333 | 0.5443 | 0.8208 | 0.9590 | 0.9959 | 1.0000 |
Reading this table tells you, for instance, that $P(X\le3)=0.8208$ directly — no addition required once the table (or calculator) has done the work.
The final entry in any cumulative binomial table is always $1$: $X$ can take no value greater than $n$, so the probability of "$X\le n$" is certain. This is a useful check that a table or calculator output is sensible.
Translating the wording
Exam wording rarely says "$P(X\le x)$" directly — it uses everyday phrases that need converting first. Since a binomial random variable only takes whole-number values, strict inequalities can always be rewritten using $\le$ or $\ge$:
| Phrase | Means | Calculate as |
|---|---|---|
| more than 5 | $X>5$ | $1-P(X\le5)$ |
| no more than 5 | $X\le5$ | $P(X\le5)$ |
| at least 5 | $X\ge5$ | $1-P(X\le4)$ |
| fewer than 5 | $X<5$ | $P(X\le4)$ |
| at most 5 | $X\le5$ | $P(X\le5)$ |
Whatever phrase a question uses, translate it into an expression built entirely from $P(X\le k)$ for some whole number $k$ — using $P(X\ge r)=1-P(X\le r-1)$ and $P(X>r)=1-P(X\le r)$ where needed. Every table and every calculator function is built around $P(X\le x)$, so this is the one form everything else should collapse into.
Worked example 1
Write the translation ("no more than 3 means $X\le3$") explicitly before calculating — it is usually worth its own mark, and prevents the easy mistake of computing $P(X\le6)$ instead of $1-P(X\le5)$ for an "at least" statement.
Worked example 2
Any "strictly between" probability, however it's phrased, can always be built from two cumulative probabilities subtracted from each other — the only care needed is deciding whether each endpoint is included, which fixes exactly which two values of $k$ to use.
Worked example 3
Rather than testing every value of $r$ from $0$, start near the distribution's mean ($np=2$ here) and move outward — this "threshold-hunting" technique reappears, almost unchanged, when finding critical regions in hypothesis testing.
$P(X<r)$ and $P(X\le r)$ are not the same for a discrete variable — $P(X<r)=P(X\le r-1)$, one step lower. Mixing these up is the single most common error in this topic, and it's always worth writing the "means" column from the translation table explicitly before calculating anything.
Check yourself
$X\sim B(12, 0.3)$. Given $P(X\le2)=0.2528$ and $P(X\le6)=0.9610$, find $P(X\ge3)$ and $P(2<X\le6)$.
(Hint: "at least 3" is the complement of "≤2".)