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Imagine that you are flipping a coin. Whether the coin lands on heads or tails does not depend on any previous flips. Each flip is an independent event. Independence in events means that the outcome of one event does not change the probability of another event happening.

In mathematical terms, two events, say Event A and Event B, are independent if and only if the probability of both events occurring together is the product of their individual probabilities. This can be written as:

• \( P(A \cap B) = P(A) \times P(B) \)

If this equation holds, then events A and B are independent. In the TB screening example, if being a recent immigrant did not affect the probability of having TB, the events "being a recent immigrant" and "having TB" would be independent.

Dependency in events is like when pulling colored marbles from a bag without replacing them; each draw changes the makeup of the bag and influences the next draw. When events are dependent, the outcome of one event affects the probability of another event occurring.

In the context of the TB screening example, the fact that the proportion of recent immigrants with TB (0.0075) is significantly higher than that of the general population (0.0006) strongly suggests dependency. When one event, such as being a recent immigrant, affects how likely another event, such as having TB, can occur, the events are dependent.

This is often observed by checking whether altering the condition of one event changes the probability of another event, demonstrating that:

• \( P(A | B) eq P(A) \)

This shows that knowing whether a kindergartner is a recent immigrant affects the probability of the kindergartner having TB.

Comparing proportions involves looking at how one percentage or fraction compares to another. It is a very useful tool in identifying relationships between different groups or features in data.

In the exercise problem, we compare the proportion of all kindergartners with TB to the proportion of recent immigrant kindergartners with TB. By comparing the figures

• 0.0006 for all kindergartners
• 0.0075 for recent immigrant kindergartners

We see that recent immigrant kindergartners are more likely to have TB. This comparison helps us determine dependency between the two events.

When comparing proportions, if there is a notable difference, it can indicate that one characteristic may have some influence over another. Thus, comparing these proportions helped us conclude that the events are dependent in the given example.