Is there a way to see a connection between Law of Total Probability and Law of Total Expectation?

Question

Both the Law of Total Probability and Law of Total Expectation are suspiciously similar looking:

$$E(X) = \sum_{i} E(X|A_i)P(A_i) $$

$$P(A) = \sum_{n} P(A|B_n)P(B_n) $$ where $B_n$ partitions the sample space.

Is one a special case of the other? Are they both just coincidentally true?

Answer 1

The second one is special case of the first. Just take $X=I_A$.

Answer 2

The probability of an event $A$ is the expected value of the corresponding indicator random variable $1_A = \cases{1 & if $A$ occurs\cr 0 & otherwise}$. Similarly for conditional probability and conditional expectation.