Intuition on joint conditional

Question

For the equation:

$p(a, b \mid c)=p(a \mid b, c) p(b \mid c)$

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I was able to prove it but can you help with some easy intuition.

$p(a,b \mid c) = p(a,b,c)/p(c) \\ p(c) =p(b,c)/p(b \mid c)$

$\therefore$

$p(a,b \mid c) =p(a,b,c)p(b \mid c)/p(b,c) \\ p(a,b \mid c) =p(b \mid c) p(a,b,c)/p(b,c) \\ p(a,b \mid c) =p(b \mid c) p(a \mid b,c)$

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Q1. Can you help me understand the intuition on previous equation.

Q2. Is there some site online that can do symbolic math on probability such as: $p(a, b \mid c)$

Answer 1

Yea the earlier answer basically said it all. More explicitly, obviously we have (because of Bayes theorem, or drawing the relevant Venn diagram, or whatever):

$$P(a,b)=P(a|b)P(b)$$

Then we can add the condition on $c$ on both sides to get

$$P(a,b|c)=P(a|b,c)P(b|c)$$

Answer 2

I think of the condition on conditional probability as "environment." Such as, rolling two dice, the numbers are independent...conditional on the dice being fair, being balanced the same, not being marked differently, etc. You just take the usual independence relation and add "given z" to every probability.