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The multiplication rule is an essential concept in probability that helps us determine the likelihood of two events occurring together. This rule is especially useful when working with dependent or conditional events. In our specific exercise, we deal with two events:

• Event L: Selecting a Latino Californian.
• Event C: The preference for life imprisonment over the death penalty amongst Californians.

By the definition of conditional probability, the multiplication rule for these two events is expressed as \( P(A \text{ and } B) = P(A) \times P(B|A) \). This means the probability of both events happening concurrently is the product of the probability of event A happening and the conditional probability of event B happening given that A has already occurred. In our problem, \( P(L \cap C) \) is calculated by multiplying the probability of someone being Latino by the probability they prefer life in prison, given they are Latino. This framework allows us to calculate the joint probability sensefully and logically.

In probability calculation, understanding how to use given values correctly is crucial. In the problem, we're tasked with finding \( P(L \cap C) \), which is the probability that a randomly selected Californian voter is both Latino and prefers life in prison without parole over the death penalty.

• First, we identified the probabilities given in the problem:
• \( P(C|L) = 0.55 \): This is the probability that a Latino Californian prefers life in prison over the death penalty.
• \( P(L) = 0.376 \): This is the probability that a person selected is Latino.

To calculate \( P(L \cap C) \), we applied the multiplication rule: \( P(L \cap C) = P(L) \cdot P(C|L) \). Substituting the known probabilities into the formula, we get \( 0.376 \times 0.55 \), which ultimately gives us \( 0.2068 \). This calculation helps us quantify the intersection of two events.

The intersection of events in probability refers to situations where we are interested in knowing the likelihood of multiple events occurring simultaneously. It is symbolized by \( \cap \), which stands for 'and'. In our scenario, we're looking at the intersection of two particular events:

• Being a Latino Californian (Event L).
• Preferring life in prison over the death penalty if convicted of first-degree murder (Event C).

When we compute \( P(L \cap C) \), we're determining the probability that both these conditions are satisfied at the same time for a randomly selected Californian. This intersection helps us focus on a specific subset of Californians, those who satisfy both conditions concurrently. Using the multiplication rule, as we did in the exercise, provides a mathematical way to calculate these intersecting probabilities efficiently. The result, 0.2068 or 20.68%, tells us about the proportion of the population that meets both criteria. This approach is useful in various fields, such as risk assessment and decision-making.