 The first line follows from the definition of conditional probability. The second follows from the fact that \(p'(T_{1\ldots9})=1\), since we’ve seen how the first \(9\) tosses go. The third line follows from an elementary theorem of the probability axioms: conjoining \(A\) with another proposition \(B\) that has probability 1 results in the same probability, i.e., \(p(A \wedge B)=p(A)\) when \(p(B)=1\). (Deriving this theorem is left as an exercise for the reader.) Finally, the last line just follows from our assumption that  \[p'(T_{10}\mid T_{1\ldots9}) = p(T_{10}\mid T_{1\ldots9})=10/11.\]   The thesis that we should generally update probabilities in this fashion is known as conditionalization. 