Total Law of Probability question

[Cyles494]

A nonsymmetric binary communication channel is shown in the figure below. Assumethe input is “0” with probability p and “1” with probability 1 − p.
Question and Solution is available in following link. I just need explanation for the steps.

https://www.cs.uic.edu/pub/ECE341s12/Homeworks/HW2_sol.pdf

Thanks

 

[HallsofIvy]

The diagram says that, if the input is 0, the output will be 0 with probability $\displaystyle 1- \epsilon_1$ and 1 with probability $\displaystyle \epsilon_1$ but if the input is 1 the output will be 0 with probability $\displaystyle \epsilon_2$ and 1 with probability $\displaystyle 1- \epsilon_2$.

We are also given that the probability the input will be 0 is p and the probability the input will be 1 is 1- p. What is the probability the output will be 0? That can happen in two ways: the input was 0 and the output was 0 or the input was 1 and the output was 0.

Given that the input was 0 what is the probability the output is 0? Multiply that by p. Given that the input was 1 what is the probability the output is 0 ? Multiply that by 1- p. Then add those two numbers.