Help with entropy calculation (binary erasure channel)

[Simfish]

So in a binary erasure channel, the book says that H(Y) [of the output] = H((1-p)(1-a),a,p(1-a)) = H(a) + (1-a)H(p), where p = Pr (X=1) [rather than X = 0] and a is the probability that ANY input of X would be erased. the problem is, how do you get from H((1-p)(1-a),a,p(1-a)) = H(a) + (1-a)H(p),? there are 3 inputs to the entropy function and they're not independent (I think) so you can't just brute force the entropy algorithm. The main thing I'm confused about is how do you get (1-a)H(p), with the (1-a) term outside of the entropy term?

 

[Evalesco]

It sounds like you’re having difficulty understanding how to use the entropy formula to calculate the output of a binary erasure channel. It can be difficult to understand at first, but the key is to remember that the entropy formula takes into account the probability of each input. In this case, we have three inputs: (1-p)(1-a), a, and p(1-a). The first two inputs are independent, so the entropy of each can be calculated separately. However, the last input is not independent, so we need to take into account the probability of each input by multiplying it by the entropy of that input. This is why we get (1-a)H(p), with the (1-a) term outside of the entropy term - because it is a factor that multiplies the entropy of the third input.