Sampling Without Replacement (VCE SSCE Mathematical Methods): Revision Notes

Sampling Without Replacement

What is sampling without replacement?

When we sample without replacement, we select items from a group and don't put them back before making the next selection. This is different from sampling with replacement, where items are returned to the group after each selection.

The key feature of sampling without replacement is that the probability of selecting a particular item changes after each selection. This happens because the composition of the group changes as items are removed.

Understanding through an example

Let's explore this concept with a practical example.

Imagine a jar containing three mints and four toffees (seven lollies in total). Bob selects two lollies from the jar without looking, and without replacing the first one before selecting the second.

Let $X$ represent the number of mints Bob selects. The random variable $X$ can take the values $0$, $1$, or $2$.

Initially, the probability that Bob selects a mint is $\frac{3}{7}$, and the probability he selects a toffee is $\frac{4}{7}$.

However, when Bob makes his second selection, only six lollies remain. The probability of selecting a mint or toffee on the second draw depends entirely on what he selected first. This dependency is the hallmark of sampling without replacement.

Method 1: Using a tree diagram

We can visualize this problem using a tree diagram, which shows all possible outcomes across the two selections.

The tree diagram branches out from the first selection to show the second selection. Notice how the probabilities on the second set of branches change depending on the first selection:

• If Bob selects a toffee first (probability $\frac{4}{7}$), then three toffees and three mints remain, giving probabilities $\frac{3}{6}$ for each on the second selection.
• If Bob selects a mint first (probability $\frac{3}{7}$), then four toffees and two mints remain, giving probabilities $\frac{4}{6}$ and $\frac{2}{6}$ respectively.

Since this involves a sequence of two dependent trials, we use the multiplication rule to find the probability of each complete outcome. We multiply along the branches:

For $X = 0$ (no mints selected, i.e., two toffees):

$$\Pr(X = 0) = \frac{4}{7} \times \frac{3}{6} = \frac{12}{42} = \frac{2}{7}$$

For $X = 1$ (one mint selected):

This can happen in two ways: toffee then mint, or mint then toffee. We add these probabilities:

$$\Pr(X = 1) = \left(\frac{4}{7} \times \frac{3}{6}\right) + \left(\frac{3}{7} \times \frac{4}{6}\right) = \frac{2}{7} + \frac{2}{7} = \frac{4}{7}$$

For $X = 2$ (two mints selected):

$$\Pr(X = 2) = \frac{3}{7} \times \frac{2}{6} = \frac{6}{42} = \frac{1}{7}$$

Method 2: Using combinations

For larger problems, drawing a complete tree diagram becomes impractical. Fortunately, we can calculate the same probabilities using combinations.

The general approach is:

$$\Pr(X = x) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$

For $X = 0$ (no mints, so two toffees):

Number of ways to select $0$ mints from $3$ available: $\binom{3}{0} = 1$

Number of ways to select $2$ toffees from $4$ available: $\binom{4}{2} = 6$

Total ways to select $2$ lollies from $7$: $\binom{7}{2} = 21$

Therefore:

$$\Pr(X = 0) = \frac{\binom{3}{0} \times \binom{4}{2}}{\binom{7}{2}} = \frac{1 \times 6}{21} = \frac{6}{21} = \frac{2}{7}$$

For $X = 1$ (one mint, one toffee):

$$\Pr(X = 1) = \frac{\binom{3}{1} \times \binom{4}{1}}{\binom{7}{2}} = \frac{3 \times 4}{21} = \frac{12}{21} = \frac{4}{7}$$

For $X = 2$ (two mints):

$$\Pr(X = 2) = \frac{\binom{3}{2} \times \binom{4}{0}}{\binom{7}{2}} = \frac{3 \times 1}{21} = \frac{3}{21} = \frac{1}{7}$$

The complete probability distribution for $X$ is:

The hypergeometric distribution

The type of probability distribution we've just explored is called the hypergeometric distribution. This distribution arises whenever we sample without replacement from a finite population containing two distinct types of items.

The hypergeometric distribution is particularly useful in quality control, ecological studies, and many other real-world applications where sampling without replacement occurs naturally.

Worked example: tagged dolphins

Remember!