Revise The binomial distribution for IB Mathematics: Analysis and Approaches (4.5)

Before any calculation, you must be sure the situation is genuinely binomial. Four conditions must all hold, and they are worth checking explicitly because a single failed condition changes the model entirely. A handy memory aid is the acronym BINS.

The condition most often overlooked is constant $p$ with independence. Sampling without replacement — drawing counters from a bag and not putting them back — changes the probability from one draw to the next, so those trials are neither independent nor identically distributed and the situation is not binomial. Sampling with replacement, or an experiment repeated under identical conditions, keeps $p$ constant and is binomial.

When a random variable $X$ counts the number of successes and satisfies the four conditions, we write it in the shorthand $X\sim B(n,p)$. This states everything at once: a binomial distribution with $n$ trials and success probability $p$. Naming the distribution this way is not decoration — in an exam it is the line that earns the first method mark, because it shows the examiner you have identified the correct model.

The probability of exactly $x$ successes in $n$ trials is $P(X=x)=\binom{n}{x}p^x(1-p)^{n-x},\qquad x=0,1,2,\dots,n,$ where $\binom{n}{x}=\dfrac{n!}{x!(n-x)!}$ counts the number of ways to arrange $x$ successes among the $n$ trials.

Read the formula as three factors: $\binom{n}{x}$ counts the arrangements, $p^x$ is the probability of $x$ successes, and $(1-p)^{n-x}$ is the probability of the remaining $n-x$ failures. The two exponents must always add to $n$; if they do not, you have mixed up $x$ and $n$. On Paper 2 you rarely evaluate this by hand — the GDC command binomPdf(n, p, x) does it — but understanding the structure is what lets you set problems up correctly.

Exam questions rarely stop at 'exactly'. They ask for ranges: 'at most 3', 'at least 10', 'fewer than 5', 'between 8 and 12'. These are cumulative probabilities, and your GDC's binomCdf function is built for them. The single most important habit is to translate the English into a probability statement before touching the calculator, because that statement is where the method mark lives and where careless slips are caught.

For any distribution we want a measure of centre (the mean) and of spread (the variance). For the binomial these come from two remarkably simple formulas that you should be able to write down instantly — they are frequently worth an easy method-and-answer pair of marks.

For $X\sim B(n,p)$: $E(X)=\mu=np,\qquad \text{Var}(X)=\sigma^2=np(1-p)=npq.$ The standard deviation is $\sigma=\sqrt{np(1-p)}$.

The mean $E(X)=np$ is the average number of successes over many repetitions. Toss a fair coin 100 times ($n=100$, $p=0.5$) and you expect $E(X)=100\times 0.5=50$ heads, with variance $\text{Var}(X)=100\times 0.5\times 0.5=25$ and standard deviation $5$. Note that the mean need not be a whole number even though $X$ itself only takes whole values — it is a long-run average, not a single outcome.

On Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) for the correct approach, or an accuracy mark (A) for the correct value that depends on that method being present. Follow-through (FT) means a wrong number early on need not cost you the marks that follow, provided your later steps are correct on your own figure. The engine also applies 'ignore subsequent working' (ISW) once a correct answer is seen, accepts equivalent forms, and accepts any answer that rounds correctly. But that protection only exists if the method is written down. Study how each mark below is earned by a specific line.

The binomial distribution is your first named probability model, and the habits it teaches carry forward. Recognising conditions before choosing a model, translating words into a precise probability statement, and using the complement for 'at least' all reappear in the normal distribution that follows, where the same GDC-and-statement discipline applies to continuous data. The mean and variance formulas foreshadow the more general expectation algebra of HL. Master the routine here — check BINS, write $X\sim B(n,p)$, state the probability, read it off, round sensibly — and the rest of the statistics topic becomes variations on a method you already own.