Given that the shop has an equal number of LED bulbs of two different types. Therefore,
Probability of Taking Type $1$ Bulb $= 0.5$
Probability of Taking Type $2$ Bulb $= 0.5$
The probability of an LED bulb lasting more than $100$ hours given that it is of Type $1$ is $0.7$, and given that it is of Type $2$ is $0.4$. i.e.,
$Prob(100+ \mid Type1) = 0.7$
$Prob(100+\mid Type 2) = 0.4$
$Prob(100+) = Prob(100+ \mid Type1) \times Prob(Type1) + Prob(100+ \mid Type2) \times Prob(Type2)$
$\quad \quad = 0.7 \times 0.5 + 0.4 \times 0.5 = 0.55.$