Revise Binomial and Poisson distributions for IB Mathematics: Applications and Interpretation (4.6)

The binomial distribution models the number of 'successes' in a fixed number of repeated trials. Picture running the same simple experiment over and over — testing a bulb, rolling a die, tossing a coin — and tallying how often the outcome you care about occurs. A situation is binomial only if it meets four conditions, conveniently remembered as BINS.

If a variable $X$ meets these conditions we write $X \sim B(n, p)$, and its probability of exactly $x$ successes is given by the formula below. On Paper 2 you will almost always evaluate it on the GDC: binomPdf(n, p, x) returns the exact probability $P(X=x)$, and binomCdf(n, p, a, b) returns the cumulative probability $P(a \le X \le b)$ across a range.

For $X \sim B(n, p)$: \n $P(X = x) = \dbinom{n}{x} p^{x} (1-p)^{n-x}, \qquad x = 0, 1, 2, \ldots, n.$ \n The mean (expected number of successes) is $E(X) = np$.

Questions rarely stop at 'exactly' and 'at most'. The phrase to watch is 'at least', and above all 'at least one'. Summing $P(X=1)+P(X=2)+\cdots$ up to $n$ is slow and error-prone; the complement is instant. Since the only alternative to 'at least one' is 'none', $P(X\ge1)=1-P(X=0)$, and for a binomial $P(X=0)=(1-p)^n$. More generally $P(X\ge k)=1-P(X\le k-1)$ — mind the '$-1$', because 'at least $k$' excludes $k-1$ but includes $k$.

The Poisson distribution models the number of times an event happens within a fixed continuous interval — of time, length, area or volume — when events arrive at a constant average rate. Think of the emails hitting your inbox in an hour, the flaws along a 10-metre roll of fabric, or the goals in a football match. There is no fixed number of trials here; the single parameter is the mean number of events in the interval, written $m$ (some books use $\lambda$).

For $X \sim Po(m)$: \n $P(X = x) = \dfrac{e^{-m} m^{x}}{x!}, \qquad x = 0, 1, 2, \ldots$ \n The mean and the variance are BOTH equal to $m$: $E(X) = \text{Var}(X) = m$. \n On the GDC use poissonPdf(m, x) for $P(X=x)$ and poissonCdf(m, a, b) for $P(a \le X \le b)$.

The equality of mean and variance is the Poisson's fingerprint. It also means that when the interval in the question is not the interval the rate was quoted for, you must scale $m$ in proportion. A rate of 6 per hour becomes a mean of 2 over 20 minutes ($6\times\tfrac13$) and a mean of 12 over 2 hours ($6\times2$). Always rescale before you calculate.

The most damaging error is applying the wrong distribution to a context — every subsequent mark then rests on a false footing. Before calculating, ask one question: am I counting successes out of a fixed number of attempts, or am I counting events arriving in a fixed interval? If you can point to a definite number of trials $n$, it is binomial; if events simply accumulate over time or space with no fixed number of trials, it is Poisson. 'How many of the 20 seeds germinate' is binomial; 'how many weeds sprout per square metre' is Poisson.

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and an accuracy mark depends on the method mark it follows. Follow-through (FT) means an error made earlier need not cost you the marks that depend on it, provided the later step is carried out correctly on your own figures. The engine also ignores subsequent working (ISW) once a correct answer appears, and accepts any equivalent form and any correctly rounded value. But that protection only exists if the method is on the page. Study how each mark below is earned by a specific line, and note that a GDC calculation is expected — you are not required to expand the formula, only to set it up correctly.

The binomial and Poisson are your first named probability models, and the habits they build carry straight into the rest of the course. The idea of a distribution with parameters, a mean read from those parameters, and probabilities computed on the GDC reappears with the normal distribution, where the mean and standard deviation set the shape of a continuous bell curve. The complement rule you used for 'at least one' is the same move that turns awkward 'at least' questions on the normal curve into quick subtractions. And the discipline that earns marks here — recognise the model, state the parameters, translate the English into an exact range, show the method — is exactly the discipline every Paper 2 probability question rewards.