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Conditional probability is a fundamental concept in probability theory. It refers to the probability of an event occurring given that another event has already occurred. This is useful when dealing with dependent events.
In our context, we have a conditional probability represented by the symbol \( P(C \mid L) \). This indicates the probability that a Latino Californian prefers life in prison without parole, knowing specifically that they are Latino.
The formula to calculate conditional probability is:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \)

However, in this situation, we are given \( P(C \mid L) = 0.55 \), meaning that 55% of Latino Californians prefer life in prison without parole over the death penalty. This helps us to analyze more intricate questions involving specific demographics.
Conditional probability helps adjust our understanding of the likelihood of an event based on new information. It's a key tool for statistical analysis and decision-making.

The union of two events in probability theory combines all outcomes that belong to either one or both of the events. It represents the scenario where at least one of the described events occurs.
In mathematical terms, if we have two events, \( A \) and \( B \), the probability of their union is denoted as \( P(A \cup B) \).
In our exercise, we needed to find \( P(L \cup C) \), meaning the probability of selecting a Californian who is either Latino or prefers life in prison without parole.
To calculate this, we used:

• \( P(L \cup C) = P(L) + P(C) - P(L \cap C) \)

This formula accounts for any overlapping probabilities, ensuring we don't double-count individuals who fit both criteria. The computed result, 0.6492, indicates a 64.92% probability that a randomly selected Californian fits at least one of these categories.

The intersection of events pertains to the probability of both events happening simultaneously. This is crucial in understanding how two conditions might overlap or co-occur in a dataset.
The intersection is symbolized as \( A \cap B \) in mathematical terms. For the given scenario, \( P(L \cap C) \) stands for the probability of selecting someone who is both Latino and prefers life in prison over the death penalty.
We calculated this using the known conditional probability, which is:

• \( P(L \cap C) = P(L) \times P(C \mid L) \)

By substituting the values, we derived \( P(L \cap C) = 0.2068 \), meaning that approximately 20.68% of the population are both Latino and prefer life in prison without parole.
Understanding intersections helps in statistical analysis, allowing insight into how different factors might correlate or interact with each other.

Statistical analysis involves using mathematical techniques to understand and interpret data. It encompasses various methods to analyze data sets and make informed decisions based on that data.
In this problem, we used probability theory as a statistical tool to evaluate and interpret the given demographic information.
Key steps in statistical analysis with probabilities include:

• Identifying known probabilities and data.
• Calculating conditional, intersection, and union probabilities.
• Interpreting these probabilities meaningfully.

By finding \( P(L \cup C) \) as 0.6492, the analysis provides a clearer picture of demographic preferences and characteristics, such as the propensity towards life in prison without parole among Latino voters.
These insights can be useful for policymakers, researchers, or anyone needing to understand or predict behaviors within a population segment.