Using the binomial distribution

The binomial distribution models a situation that is repeated a fixed number of times, where each repetition (a 'trial') has just two outcomes, a 'success' or a 'failure', and the probability of success stays the same every time. Tossing a coin 10 times and counting heads, or checking 5 items and counting how many are faulty, are classic examples.

If $X$ is the number of successes, then $X$ follows a binomial distribution written $X\sim B(n,p)$, where $n$ is the number of trials and $p$ is the probability of success in one trial. The formula $P(X=r)={}^{n}C_{r}\,p^{r}(1-p)^{n-r}$ then gives the probability of exactly $r$ successes.

This lets us answer questions such as 'exactly 2', 'at least 3', or 'at most 1' success, and to find the mean $np$ and variance $np(1-p)$.

Reach for the binomial distribution when a question describes a fixed number of repeated trials and asks for the probability of a certain number of 'successes'. Look for four features: a set number of trials $n$; each trial has only two outcomes; the trials are independent; and the success probability $p$ is the same each time.

Signal phrases include 'in $n$ trials', 'each with probability $p$', 'exactly', 'at least', or 'at most'. If any trial affects the next, for example drawing without replacement so the probability changes, it is not binomial.

When the variable is instead a continuous measurement such as height or mass, you are looking at the normal distribution rather than the binomial.

1. 1

   Check the conditions

   Fixed $n$ trials, two outcomes per trial, independent trials, and a constant success probability $p$.

2. 2

   Identify n, p and r

   State the number of trials $n$, the success probability $p$, and the number of successes $r$ you need.

3. 3

   Write the formula

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

4. 4

   Substitute the values

   Put $n$, $p$, $1-p$ and $r$ into the formula; check the two powers add up to $n$.

5. 5

   Evaluate

   Work out ${}^{n}C_{r}$, then the two powers, then multiply the three parts together.

6. 6

   Combine for 'at least' or 'at most'

   Add the separate terms you need, or use the complement, e.g. $P(X\ge 1)=1-P(X=0)$.

• Swapping $p$ and $1-p$, or being unclear about which outcome counts as a 'success'.
• Using the wrong $r$, or writing exponents that do not add up to $n$, the powers of $p$ and $1-p$ must total $n$.
• For 'at least one', calculating a long sum instead of using the complement $1-P(X=0)$.
• Confusing the mean $np$ with the variance $np(1-p)$, or taking the variance as the standard deviation.
• Rounding ${}^{n}C_{r}$ or the powers too early, so the final probability drifts off.

The step students most often get wrong is the very first one: checking that the situation really is binomial, and being clear about which outcome is the 'success' that $p$ belongs to. In a one-to-one lesson our teachers ask you to say the four conditions aloud and name $n$, $p$ and $r$ before any calculating begins, so the setup is right every time.

We also drill the complement shortcut for 'at least' questions until it feels natural. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we teach the binomial distribution, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.