Conditional probability from XYZ table

Question

This question was migrated from Stack Overflow because it can be answered on Cross Validated. Migrated 14 hours ago .

Given this table, is X conditionally independent of Y give Z?

I know conditional independence is P(X|Y,Z) = P(X, Z), but I'm a bit confused about how I can use that with values from the table. I understand how to get P(X|Y,Z) from the table, but am a bit confused about the right hand side of the equation, for example P(X = 0, Z = 0) can have 2 values? I think I'm missing something.

Answer 1

It's a self-study question, so let me give you some hints so that you could solve it by yourself.

• The table shows the joint probabilities $\\sum_{x,y,z} P(x, y, x) = 1$ .

• Recall the definition of conditional probability

  $$ P(A|B) = \\frac{P(A \\cap B)}{P(B)} $$

  it will be needed to compute the conditional probabilities from the joint probabilities.

• You also need the law of total probability

  $$ P(A) = \\sum_n P(A \\cap B_n) $$

  to get the marginal probabilities.

• Use the above and check against the definition of conditional independence

  $$ P(A|B,C) = P(A|C) $$

  [...] In this case, $A$ and $B$ are said to be conditionally independent given $C$