For "The Glimmer" the bulbs are connected in parallel. In that case the set works (lights) as long as at least one bulb functions. (The company says that the set "works (lights)" when the set is not completely dark.)
Suppose that one uses $n$ light bulbs to construct a set, and that the light bulbs life lengths $T_1, T_2, \ldots, T_n$ (unit days) are independent and $\operatorname{Exp}(1)$ distributed. Let $T_G$ the life length for "The Glimmer".
(a) How large does $n$ have to be to have $P\left(T_G>10\right)>0.5$ ?
My understanding is that since $T_n$ are all independent so $T_G>10$ means any of the $T_n>10$
\begin{aligned} P\left(T_G>10\right) &=P\left(\left(T_1>10\right) \cup \cdots \cup\left(T_n>10\right)\right) \\ &=P\left(T_1>10\right)+\cdots +P\left(T_n>10\right) \\ &=0.00005 n>0.5 \\ n &>10000 \end{aligned}
But this is different than the solution given,
\begin{aligned} P\left(T_G>10\right) &=1-P\left(T_G \leqslant 10\right) \\ &=1-P\left(T_1 \leqslant 10, \cdots, T_n \leqslant 10\right) \\ &=1-\left(F_{T_1}(10)\right)^n \\ n &>15267.2 \end{aligned}
I don't understand why in the solution it used intersection of $T_n$ instead of unions of $T_n$. Thank you in advance!
Your second equality is actually an inequality. They are equal only when all the events are disjoint.
Notice their work uses the independence of the events. At least one $T_i$ occurring is the same as all $T_i$s not occurring.