https://file.aiquickdraw.com/mathbot/file/17265023758056z50oir3.jpg

The image is now visible. I'll proceed by analyzing and solving the provided probability problems.

The question describes three events:
- \( A \): The person is sick.
- \( B \): The person's blood test is positive.
- \( C \): The person has no symptoms.

We're given several conditional probabilities and requested to calculate the following:

### Known Probabilities:
- \( P(A) = 0.3 \)
- \( P(B | A) = 0.75 \)
- \( P(B | A^c) = 0.20 \)
- \( P(C | A \cap B) = 0.20 \)
- \( P(C | A^c \cap B) = 0.15 \)
- \( P(C | A \cap B^c) = 0.80 \)
- \( P(C | A^c \cap B^c) = 0.90 \)

### Required Calculations:
(a) \( P(A \cap B \cap B^c \cap C) \)

(b) \( P(B^c \cup C) \)

(c) \( P(C) \)

(d) The probability that the person is sick, given that they are asymptomatic and the test result is negative.

I'll now solve these step by step.It seems like I can't do more advanced data analysis right now. Please try again later.

If you have any questions about the problem setup or need help understanding the approach to solving these probabilities, feel free to ask! Would you like me to guide you through the solution manually, or attempt another task?

Here are some related questions to expand on the topic:
1. How is conditional probability applied in Bayesian analysis?
2. What is the significance of the complement rule in probability?
3. How can you calculate joint probability using the multiplication rule?
4. How does the law of total probability work in practical scenarios?
5. What are some common uses of Bayes' Theorem in real-life situations?

**Tip:** When dealing with multiple events, using a probability tree diagram can help visually organize the relationships and conditional probabilities.

Considere os seguintes eventos: A = {A pessoa está doente} B = {O teste de sangue da pessoa resultou positivo para a doença.} C = {A pessoa não apresenta sintomas.} Assuma agora que P(A) = 0.3, P(B|A) = 0.75, P(B|A^c) = 0.20, P(C|A ∩ B) = 0.20, P(C|A^c ∩ B) = 0.15, P(C|A ∩ B^c) = 0.80, P(C|A^c ∩ B^c) = 0.90. Calcule: (a) P(A ∩ B ∩ B^c ∩ C); (b) P(B^c ∪ C); (c) P(C); (d) a probabilidade de a pessoa estar doente, dado que é assintomática e o teste tenha resultado negativo.

Explore the application of conditional probability and Bayes' Theorem in a medical testing scenario. Learn how to calculate joint probabilities, complements, and conditional probabilities with given data points such as P(A) = 0.3, P(B|A) = 0.75, and more. Suitable for students studying probability and statistics.

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