$($Total: $10$ points$)$ In a bakery, three ovens A$,$ B$,$ and C are used to bake cookies. Oven A is responsible for $30\%$ of the bakery's total cookie production, oven B for $50\%,$ and oven $C$ for $20\%.$ The quality control checks reveal that $5\%$ of the cookies from oven A$,4\%$ of the cookies from oven B$,$ and $2\%$ of the cookies from oven C are burnt. It's also found that $10\%$ of the cookies from oven A$,6\%$ of the cookies from oven B$,$ and $5\%$ of the cookies from oven $C$ are underbaked.

a$.$ If a cookie is selected at random and found to be burnt, what is the probability that it was baked in oven B$?(5$ points$)$