H2 Maths 6.2 - When Can You Use a Binomial Model?

Step 1 - The four conditions

The binomial distribution models the number of successes in repeated trials.

Step 1 - The four conditions

But you can only use it when four conditions are met.

Step 1 - The four conditions

Condition one: fixed number of trials, n.

Step 1 - The four conditions

Condition two: each trial has exactly two outcomes, success or failure.

Step 1 - The four conditions

Condition three: the probability of success, p, is constant across all trials.

Step 1 - The four conditions

Condition four: the trials are independent.

Step 1 - The four conditions

In exams, you must state all four conditions to earn full marks.

Step 2 - Example that works

A factory tests fifteen light bulbs. Each bulb independently passes or fails with a constant defect rate of zero point zero five.

Step 2 - Example that works

Fixed n? Yes, fifteen bulbs.

Step 2 - Example that works

Two outcomes? Yes, pass or fail.

Step 2 - Example that works

Constant p? Yes, zero point zero five for every bulb.

Step 2 - Example that works

Independent? Yes, each bulb is tested separately.

Step 2 - Example that works

All four conditions hold. X, the number of defective bulbs, follows Bin fifteen comma zero point zero five.

Step 3 - Example that fails

Now consider drawing three cards from a standard deck without replacement. You want to count the number of aces.

Step 3 - Example that fails

Fixed n? Yes, three draws.

Step 3 - Example that fails

Two outcomes? Yes, ace or not ace.

Step 3 - Example that fails

But constant p? No.

Step 3 - Example that fails

The first draw has probability four over fifty-two. If you draw an ace, the second draw has probability three over fifty-one. The probability changes.

Step 3 - Example that fails

And because p changes, the trials are also not independent.

Step 3 - Example that fails

Binomial does not apply here. This scenario would use the hypergeometric distribution instead.

Step 4 - Exam strategy

In the exam, follow this routine.

Step 4 - Exam strategy

First, identify the trial and what counts as success.

Step 4 - Exam strategy

Second, check each of the four conditions explicitly.

Step 4 - Exam strategy

Third, if any condition fails, state which one and why.

Step 4 - Exam strategy

The most common condition that fails is constant p, usually because of sampling without replacement or changing conditions.

Step 4 - Exam strategy

Always write the justification in words. Do not just write X tilde Bin n p without explaining why.