Understanding Probability of Inclusion in SRSWR

This question asks about the probability that a specific individual or item from a population is included in a sample when using Simple Random Sampling With Replacement (SRSWR). Let's break down what this means and how to calculate the probability.

What is Simple Random Sampling With Replacement (SRSWR)?

• Population: We start with a population of a known size, denoted by \(N\).
• Sampling: We draw items one by one to form our sample.
• Simple Random: At each draw, every member of the population has an equal chance of being selected. The probability of selecting any specific member is \(\dfrac{1}{N}\).
• With Replacement: After an item is selected, it is returned to the population before the next item is drawn. This means the population size remains \(N\) for every single draw, and the draws are independent events.
• Sample Size: We draw a total of \(n\) items to form our sample. Note that with replacement, the same item can be selected multiple times in the same sample.

Calculating the Probability of Inclusion for a Specified Member

We are interested in the probability that a particular, specified member of the population is part of the final sample of size \(n\).

It's often easier to calculate the probability of the opposite event – that the specified member is not included in the sample at all – and then subtract that probability from 1.

Let's consider a single draw:

• The probability that the specified member is selected in a single draw is \(\dfrac{1}{N}\).
• The probability that the specified member is not selected in a single draw is \(1 - \dfrac{1}{N} = \dfrac{N-1}{N}\).

Now, let's consider the sample of size \(n\) drawn with replacement. Since each draw is independent, the probability that the specified member is not selected in *any* of the \(n\) draws is the product of the probabilities of not selecting it in each individual draw.

• Probability of not selecting the specified member in the 1st draw: \(\dfrac{N-1}{N}\)
• Probability of not selecting the specified member in the 2nd draw: \(\dfrac{N-1}{N}\)
• ...
• Probability of not selecting the specified member in the \(n\)-th draw: \(\dfrac{N-1}{N}\)

So, the probability that the specified member is not included in the sample of size \(n\) is:

\[ P(\text{Specified member not included}) = \left(\dfrac{N-1}{N}\right) \times \left(\dfrac{N-1}{N}\right) \times \cdots \times \left(\dfrac{N-1}{N}\right) \quad (\text{n times}) \]

\[ P(\text{Specified member not included}) = \left(\dfrac{N-1}{N}\right)^n \]

This can also be written as:

\[ P(\text{Specified member not included}) = \left(1 - \dfrac{1}{N}\right)^n \]

Finally, the probability that the specified member is included in the sample of size \(n\) is the complement of the above probability:

\[ P(\text{Specified member included}) = 1 - P(\text{Specified member not included}) \]

\[ P(\text{Specified member included}) = 1 - \left(1 - \dfrac{1}{N}\right)^n \]

Comparing with the Options

Let's compare our derived probability with the given options:

• Option 1: \(\dfrac{1}{N}\) (Probability of selection in a single draw) - Incorrect.
• Option 2: \(\dfrac{n}{N}\) (Not the correct probability for inclusion in SRSWR) - Incorrect.
• Option 3: \(1-\left(1-\dfrac{1}{N}\right)^n\) (Our calculated probability) - Correct.
• Option 4: \(\dfrac{(N-1)^n}{N^n} = \left(\dfrac{N-1}{N}\right)^n = \left(1-\dfrac{1}{N}\right)^n\) (Probability of *not* being included) - Incorrect.

The probability that a specified member is included in a sample of size \(n\) drawn under SRSWR from a population of size \(N\) is \(1 - \left(1 - \dfrac{1}{N}\right)^n\).

Revision Table: SRSWR Probability Concepts

Additional Information: SRS vs SRSWR Inclusion Probability

It's important to distinguish SRSWR from Simple Random Sampling Without Replacement (SRSWOR).

• SRSWR: As calculated above, the probability of inclusion for a specified member is \(1 - \left(1 - \dfrac{1}{N}\right)^n\). Each draw is independent.
• SRSWOR: In SRSWOR, once an item is selected, it is not returned. For a sample of size \(n\), the probability that a specified member is included is simply \(\dfrac{n}{N}\) (assuming \(n \le N\)). This is because there are \(N\) items in total, and we select \(n\) unique items. The probability that any specific item is one of the \(n\) selected is the ratio of sample size to population size.

The inclusion probabilities are different because the sampling methods treat the population differently after each selection.