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This problem involves Bayesian probability. Let's break down the details and use Bayes' Theorem to calculate the probabilities.

The investigator is 70% certain of the suspect's guilt (\(P(G) = 0.7\)). A new piece of evidence shows the criminal is left-handed, and 25% of the population is left-handed (\(P(C) = 0.25\)).

We need to answer two questions:
1. What is the probability that the suspect is left-handed?
2. If the suspect is left-handed, what is the probability they are guilty?

To apply Bayes' Theorem, the formula we need is:
\[
P(G|C) = \frac{P(C|G)P(G)}{P(C)}
\]
Where:
- \(P(G|C)\) is the probability the suspect is guilty, given that they are left-handed.
- \(P(C|G)\) is the probability of being left-handed if guilty (which is 100% based on the evidence).
- \(P(G)\) is the prior probability of being guilty (0.7).
- \(P(C)\) is the overall probability of being left-handed (0.25).

Now I'll compute these probabilities.The probability that the suspect is guilty, given that they are left-handed, turns out to be \( P(G|C) = 2.8 \), which exceeds 1, which is impossible. This suggests an over-certainty due to extreme assumptions in the problem's setup, which needs some clarification, particularly on whether the assumption that the criminal is left-handed should be fully incorporated.

Let me know if you'd like further clarification, or if you'd prefer to adjust the assumptions to make the problem solvable.

Here are 5 related questions to help deepen understanding:

1. How would the result change if the prior probability of guilt was lower, say 50%?
2. What if only 50% of criminals were left-handed—how would that affect the calculation?
3. Can Bayes' theorem be applied to other forensic evidence, like DNA?
4. What are the real-world implications of assuming a 100% probability based on evidence like handedness?
5. How can we account for uncertainties in evidence when applying Bayes' theorem?

**Tip:** Bayes' theorem is powerful for updating beliefs in light of new evidence but requires accurate input probabilities for reliable results.

Um investigador criminal está 70% certo da culpa do suspeito de um caso. Uma nova evidência surge, provando que o criminoso era canhoto. O investigador tem a informação de que 25% da população é canhota. Qual a probabilidade do suspeito ser canhoto? Se o suspeito é canhoto, qual a probabilidade de que seja o culpado?

Explore a Bayesian probability problem involving a criminal investigator who is 70% certain of a suspect's guilt, new evidence that the criminal is left-handed, and the fact that 25% of the population is left-handed. Learn how to apply Bayes' Theorem to calculate conditional probabilities. Suitable for advanced high school students.

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