Exponential bulb outliving ten bulbs used in sequence

Ten independent bulbs have lifetimes $T_1,\dots,T_{10}$, each exponential with rate $\tfrac1{10}$. An eleventh bulb, independent of them, has lifetime $X$ that is exponential with rate $\tfrac1{90}$. The probability $\mathbb{P}(X>T_1+T_2+\cdots+T_{10})$ equals $q^b$ with $q\in(0,1)$ rational and $b$ the largest possible integer. Find the product $qb$.

$9$. By memorylessness, each time a short-lived bulb fails the eleventh bulb starts afresh, so the event is ten independent wins of $X$ over a single $T_i$. Each has probability $\tfrac{1/10}{1/10+1/90}=\tfrac{9}{10}$, giving $(\tfrac{9}{10})^{10}$, so $q=\tfrac{9}{10}$, $b=10$ and $qb=9$.