 The task is to justify taking the first route rather than the second: sticking to our conditional belief that, if \(T_{1\ldots9}\), then \(T_{10}\) has probability \(10/11\), even once we’ve learned that indeed \(T_{1\ldots9}\). Standing by one’s conditional probabilities in this way is known as “conditionalizing”, because one thereby turns the old conditional probabilities into new, unconditional probabilities. To see why sticking by your old conditional probabilities amounts to turning them into unconditional probabilities, let’s keep using \(p\) to represent the prior probabilities, and let’s introduce \(p'\) to stand for the new, posterior probabilities after we learn that \(T_{1\ldots9}\). If we stand by our prior conditional probabilities, then   \[p'(T_{10}\mid T_{1\ldots9}) = p(T_{10}\mid T_{1\ldots9})=10/11.\]   And since we now know that \(T_{1\ldots9}\), \(p'(T_{1\ldots9})=1\). It then follows that \(p'(T_{10})=10/11\):