Mary is conducting research on Covid-19 and its possible symptoms. The study consists of 3 random variables: A, B and C, which are defined as follows:

1. A: Loss of smell

2. B: Presence of Covid -19

3. C: Loss of taste

By analyzing publicly available data, she is able to get full joint probability distribution table of the 3 variables, as shown in Table 3. She wants to know the relationships between A, B, C and asks for your help for this task. First, she wants to know whether A and C are independent, given B. In order to do that, you should calculate the following probabilities, assuming B = True. Answer the following questions (round to 6 decimal places):

• What is the value of P(A|B,C)?
• What is the value of P(A|B, C)?
• What is the value of P(C|B, A)?
• What is the value of P(C|B, A)?

You report your findings to her. But she is not convinced. She asks, what if B = False? Assuming B = False, answer the following questions (round to 6 decimal places):

• What is the value of P(A| B,C)?
• What is the value of P(A| B, C)?
• What is the value of P(C| B, A)? Type Your Answer Here: (h) (1.5 points)
• What is the value of P(C| B, A)?
• Based on this, are A and C conditionally independent, given B? True False Then, she asks: Does the relationship between A and C change if B is unknown?
• What is the value of P(A|C)?
• What is the value of P(A| C)?
• What is the value of P(C|A)?
• What is the value of P(C| A)?
• Based on this, is A independent of C, if B is unknown?