Understanding the binomial and normal distributions

This chapter introduces two probability distributions, and the first skill is simply telling them apart. The question that decides it is: are you counting something, or measuring something?

If you are counting how many times an event happens in a fixed number of tries, how many of 10 seeds germinate, how many of 5 free throws go in, you are in the world of the binomial distribution, which is discrete (whole-number outcomes). If you are measuring a quantity that can take any value on a scale, and that value tends to cluster around an average, heights, masses, lifetimes, you are in the world of the normal distribution, which is continuous.

The binomial applies when four conditions hold: there is a fixed number of trials $n$; each trial has only two outcomes (call them success and failure); the probability of success $p$ is the same every trial; and the trials are independent. When that setup fits, the probability of getting exactly $r$ successes is:

$$P(X=r)={}^{n}C_{r}\,p^{r}(1-p)^{n-r}$$

Each piece has a meaning. The $p^{r}$ is the chance of $r$ successes happening; the $(1-p)^{n-r}$ is the chance of the remaining trials being failures; and the ${}^{n}C_{r}$ counts how many different orders those successes could occur in.

As a worked example, suppose a trial succeeds with probability $p=0.2$ and you run $n=5$ trials. The probability of exactly 2 successes is ${}^{5}C_{2}(0.2)^{2}(0.8)^{3}=10\times 0.04\times 0.512=0.2048$.

Two summary numbers are worth knowing. On average, the number of successes is the mean $np$, and the spread is measured by the variance $np(1-p)$.

For our example, the mean is $5\times 0.2=1$ success, and the variance is $5\times 0.2\times 0.8=0.8$. These are quick to state and often asked directly.

The normal distribution is the familiar symmetric bell curve. It is described by two numbers: the mean $\mu$, which sits at the centre and peak, and the standard deviation $\sigma$, which controls how wide the curve spreads.

Because the variable is continuous, we never ask for the probability of an exact value; we ask for the probability that it falls in a range, for instance, that a randomly chosen item measures more than a certain amount. That probability is the area under the curve over that range.

The engine that makes this workable is standardisation. Any normal variable $X$ is converted into the standard normal variable $Z$, which has mean 0 and standard deviation 1, using:

$$Z=\frac{X-\mu}{\sigma}$$

The $Z$-value tells you how many standard deviations $X$ lies from the mean. Suppose a quantity is normally distributed with mean $\mu=60$ and standard deviation $\sigma=5$, and you want the probability it exceeds 70.

Standardise: $Z=\frac{70-60}{5}=2$. So the question becomes $P(Z>2)$, which you read from the standard normal table (or the calculator's distribution function).

The table gives the area, and that area is your probability. Every normal problem reduces to this: standardise, then read the area.

Most lost marks here come from a small number of predictable errors:

• Confusing standard deviation with variance. Standardising uses $\sigma$, the standard deviation; if the question gives you the variance, take its square root first. Mixing these up is the single most common normal-distribution error.
• Muddling probability directions. A table may give the area to the left, $P(Z<z)$, or to the right. Sketch the bell curve, shade the region you actually want, and use symmetry ($P(Z<-z)=P(Z>z)$) rather than guessing.
• Using the binomial when trials are not independent or $p$ changes. If you draw items without replacement, $p$ shifts each time, so the binomial no longer applies exactly.
• Forgetting that mean $=np$ and variance $=np(1-p)$ are for the binomial, not the normal. Keep the two distributions' formulas in separate mental boxes.

A steadying thought: because SPM Add Math is marked analytically, you gain method marks for the correct setup, writing the binomial expression with the right $n$, $r$, and $p$, or standardising with the correct $Z$, even if the final decimal is slightly off. A clear sketch of the bell curve, with the wanted area shaded, is genuinely worth marks and prevents direction errors at the same time.

This chapter rewards a clear head more than raw speed: choose the right distribution, keep standard deviation and variance straight, and sketch the area you are after. When a student is stuck, it is almost always one of those three, and a teacher watching your working can pinpoint which in a single problem, then rebuild it until it feels automatic.

Our teachers are experienced, and lessons are online and taught in English, which also helps with the English terms, while SPM papers are set in both Bahasa Melayu and English.

If you would like to try a session, the first lesson is a one-hour paid class at the teacher's rate, from RM50 an hour, depending on the teacher's experience, quoted on WhatsApp. We start from where you are, whether that is the binomial formula or reading the normal table, and build outward at your own pace, without pressure.