$(12$ marks$)$ A bin contains $3$ types of flashlights. The probability that a type $1$ flashlight will give more than $100$ hours of use is $0\mathrm{.}6,$ with the corresponding probabilities for type $2$ and type $3$ flashlights being $0\mathrm{.}4$ and $0\mathrm{.}5,$ respectively. Suppose that $30\%$ of the flashlights in the bin are type $1,30\%$ are type $2,$ and $40\%$ are type $3.$

$(1)$ What is the probability that a randomly chosen flashlight will give more than $100$ hours of use?

$(2)$ Given that a flashlight lasted more than $100$ hours, what is the probability that it was a type $1$ flashlight?