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5. [12 pts] Inference Using Full Joint Distributions Consider three Boolean random variables A, B, and C. We abbreviate the propositions A = true, B = true, C = true as a, b, c, respectively, and A = false, B = false, C = false as ¬a, ¬b, ¬c, respectively. Given below is a table showing their joint probability distributions. | | b | | ¬b | | | :--- | :--- | :--- | :--- | :--- | | | c | ¬c | c | ¬c | | a | 0.1 | 0.15 | 0.1 | 0.2 | | ¬a | 0.3 | 0.05 | 0.05 | 0.05 | (a) [4 pts] Compute the conditional probability distribution P(A | ¬c) = ?P(a | ¬c), P(¬a | ¬c)?. (To normalize a vector, say, ?x, y?, you may write ??x, y?, which represents ?x/(x+y), y/(x+y)?.) (b) [4 pts] Compute the conditional probability distribution P(B | ¬c) = ?P(b | ¬c), P(¬b | ¬c)?. (c) [4 pts] Are A and B conditionally independent given C? Explain why or why not.

          5. [12 pts] Inference Using Full Joint Distributions

Consider three Boolean random variables A, B, and C. We abbreviate the propositions A = true, B = true, C = true as a, b, c, respectively, and A = false, B = false, C = false as ¬a, ¬b, ¬c, respectively. Given below is a table showing their joint probability distributions.

| | b | | ¬b | |
| :--- | :--- | :--- | :--- | :--- |
| | c | ¬c | c | ¬c |
| a | 0.1 | 0.15 | 0.1 | 0.2 |
| ¬a | 0.3 | 0.05 | 0.05 | 0.05 |

(a) [4 pts] Compute the conditional probability distribution P(A | ¬c) = ?P(a | ¬c), P(¬a | ¬c)?. (To normalize a vector, say, ?x, y?, you may write ??x, y?, which represents ?x/(x+y), y/(x+y)?.)

(b) [4 pts] Compute the conditional probability distribution P(B | ¬c) = ?P(b | ¬c), P(¬b | ¬c)?.

(c) [4 pts] Are A and B conditionally independent given C? Explain why or why not.

5. [12 pts] Inference Using Full Joint Distributions

Consider three Boolean random variables A, B, and C. We abbreviate the propositions A = true, B = true, C = true as a, b, c, respectively, and A = false, B = false, C = false as ¬a, ¬b, ¬c, respectively. Given below is a table showing their joint probability distributions.

| | b | | ¬b | |
| :— | :— | :— | :— | :— |
| | c | ¬c | c | ¬c |
| a | 0.1 | 0.15 | 0.1 | 0.2 |
| ¬a | 0.3 | 0.05 | 0.05 | 0.05 |

(a) [4 pts] Compute the conditional probability distribution P(A | ¬c) = ?P(a | ¬c), P(¬a | ¬c)?. (To normalize a vector, say, ?x, y?, you may write ??x, y?, which represents ?x/(x+y), y/(x+y)?.)

(b) [4 pts] Compute the conditional probability distribution P(B | ¬c) = ?P(b | ¬c), P(¬b | ¬c)?.

(c) [4 pts] Are A and B conditionally independent given C? Explain why or why not.

Added by Javier A.

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