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Joint probability is the measure of two events happening at the same time. It helps us understand the likelihood of two or more events occurring together. In the given exercise, we want to find the probability that a randomly selected resident is both female and a smoker.

Remember the formula: \( P(A \cap B) \) which is the joint probability of events A and B. Here, we have A being 'female' and B being 'smoker'.

Using the conditional probability formula: \( P(F \cap S) = P(F|S) \times P(S) \)
The term \( P(F|S) \) represents the probability of being a female given that the person smokes. \( P(S) \) is the probability of being a smoker.

Substitute the given values:

• Probability of being a smoker \( P(S) = 0.203 \)
• Probability of being a female given that the person smokes \( P(F|S) = 0.445 \)

So, we calculate: \[ P(F \cap S) = 0.445 \times 0.203 \approx 0.090335 \]

This tells us there is approximately a 9.03% chance that a randomly selected resident is both female and a smoker.

Understanding the likelihood of an individual smoking can offer valuable health and behavioral insights. In this exercise, we're given the probability that a person smokes, 0.203 or 20.3%. This means that out of every 100 individuals, roughly 20 to 21 individuals are expected to be smokers.

If we were to visualize this probability, think of it like drawing 20 smokers out of a pool of 100 people at random. It’s a relatively straightforward concept but foundational to understanding more complex probabilities.

When dealing with probability, always remember:

• Probability is a measure of how likely an event is to occur.
• The probability value ranges between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
• A probability of 0.203 means there is a 20.3% chance of the event occurring.

This provides a basis for calculating joint probabilities, like finding the probability of being both a female and a smoker in the exercise.

In statistics, an event is considered unusual if its probability is less than 0.05 (or 5%). This criterion helps us identify occurrences that are rare or exceptional.

For example, if an event happens less than 5 times out of 100, it’s deemed unusual. This helps in making decisions and drawing conclusions based on how common or rare an event is.

In our exercise, we calculated the joint probability of a resident being female and a smoker to be approximately 0.090335 (or 9.03%).

Since 0.090335 is greater than 0.05, this event is not considered unusual. It suggests that encountering a female smoker in the population is relatively common and not a rare occurrence.

The unusual events criterion is a handy tool to quickly judge the commonality of an occurrence without needing complex analysis. This criterion answers questions like:

• Is it rare to find someone with these characteristics?
• Should this finding be surprising?

By comparing the event probability with the 0.05 threshold, we can easily determine whether an event is unusual or not.