<image> A company randomly selects $$100$$ products from a production line and measures a quality index value $$x$$ for these products. The frequency distribution histogram of the measurement results is shown in the figure.
$$(1)$$ Find the sample mean $$ \overline {x}$$ of the quality index values of these $$100$$ products $$(\text{data within the same group is represented by the midpoint value of the interval of that group})$$; 
$$(2)$$ If the quality index value $$x > \overline {x}$$, the product is classified as a superior product $$(\text{the frequency of $$x$$ falling within each interval is taken as the probability of $$x$$ taking that interval value}).$$ The quality inspection department conducts a quality inspection of the products on this production line. The inspection plan is: First, randomly select $$3$$ products from this batch for inspection. If all $$3$$ are superior products, the batch passes the inspection; if only $$2$$ are superior products, then randomly select $$1$$ more product from this batch for inspection. If it is a superior product, the batch passes the inspection; in other cases, the batch fails the inspection. The independence of each product being qualified or not is assumed.
$$(i)$$ Find the probability that this batch of products passes the inspection; 
$$(ii)$$ It is known that the inspection cost per product is $$100$$ yuan, and every selected product needs to be inspected. Denote the total inspection cost for this batch of products by $$Y$$ $$(\text{unit: yuan})$$. Find the distribution and the mathematical expectation of $$Y$$.