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The Law of Total Probability is a fundamental concept in probability theory. It provides a way to calculate the probability of an outcome by considering all possible scenarios that can lead to that outcome. In this case, we want to determine the probability, \( P(C) \), that a Californian prefers life in prison without parole over the death penalty. To apply the law, we first divide the scenarios into two mutually exclusive events: the person is a Latino \( (L) \) or not a Latino \( (L') \). Each scenario contributes to the total probability proportionally based on its own likelihood and the conditioned probability of the outcome in that scenario.The formula is:\[ P(C) = P(C | L)P(L) + P(C | L')P(L') \]Where:- \( P(C | L) \) is the probability that a Latino prefers life in prison without parole.- \( P(C | L') \) is the similar probability for non-Latinos.- \( P(L) \) and \( P(L') \) are the probabilities of the voter being Latino or not, respectively.This approach ensures that all possible pathways contributing to preferring life in prison are considered, resulting in a comprehensive probability calculation.

Complementary Probability is a simple yet crucial concept in probability. It represents the idea that the probability of an event not occurring is equal to the total probability minus the probability of the event occurring.To use complementary probability in our problem, we need to find \( P(L') \), the probability that a Californian voter is not Latino. Since only two possibilities exist—being Latino or not being Latino—their probabilities must sum to 1:\[ P(L') = 1 - P(L) \]In this example:- \( P(L) = 0.376 \)So, \( P(L') = 1 - 0.376 = 0.624 \)This step is vital for accurately applying the Law of Total Probability, allowing us to incorporate both Latino and non-Latino scenarios effectively.

Probability Calculation involves determining the likelihood of a certain outcome by appropriately applying probabilities for different conditions. Here, after defining our relevant probabilities and events, we sum up the contributions from different groups based on their conditional probabilities.For Latino voters:- \( P(C | L) = 0.55 \) and \( P(L) = 0.376 \)- So, \( P(C | L) \times P(L) = 0.55 \times 0.376 = 0.2068 \)For Non-Latino voters:- We use the overall probability \( P(C) = 0.48 \)- Then solve for \( P(C | L') \) using \( P(C) = 0.48 \) in the formula:\[ 0.48 = 0.2068 + P(C | L') \times 0.624 \]- Rearrange to find:\( P(C | L') = \frac{0.2732}{0.624} \approx 0.4385 \)Combining both:- Verify by computing \( 0.4385 \times 0.624 \approx 0.2734 \)- Adding up contributions: \( 0.2068 + 0.2734 \approx 0.48 \)The result confirms the probability \( P(C) = 0.48 \) is correctly calculated, encompassing all registered voters appropriately.