 Stock-numeraire measure in a binomial market

Let $Q$ be the risk-neutral measure in an $N$-period binomial market. The stock-numeraire measure is defined on terminal events byIf the $Q$-probability of an up move is $q$, its up probability under $\hat{Q}$ isThis change of measure converts discounted expectations containing a factor $S_N$ into probabilities under $\hat{Q}$.