I am currently reading Probability and Computing: Randomized Algorithms and Probabilistic Analysis and have encountered this passage.
Now my question lies in the fact that Pr(B|E1) = Pr(B & E1) / Pr(E1). However within the textbook, they only seem to be calculating Pr(B|E1) (I have taken the example of E1 here and this is the same case with E2 and E3).
Given you know which coin is biased, the coin tosses are independent; so $$P(B|E_1)=P(H_1|E_1)\cdot P(H_2|E_1)\cdot P(T_3|E_1)=\frac{2}{3}\cdot \frac{1}{2}\cdot \frac{1}{2}$$ where event $H_1$ = first coin landed on heads, etc.