[Darmani]

You can write P(cold outside | do(turned on heat=X)), where X is another random variable. So P(cold outside | do(turned in heat=X=1)) will be equal to P(cold outside | do(turned on heat=1))P(X=1 | cold outside) = P(cold outside)P(X=1 | cold outside).

But, you might want to consider making "turned on heat" part of the system in this case, and go back to using the classic conditioning operator instead of the do operator.

This is covered in chapter 4 of Pearl's Causality.