Question $3$B $-$ Bayesian Learning $[24\%]$

Consider a quality control problem in a manufacturing plant. There are two alternative hypotheses: $(1)$ that a manufactured product is defective, and $(2)$ that the product is not defective. The available data is from a product testing process with two possible outcomes: $+($positive$)$ and $-($negative$).$

We have the following probabilities:

Prior Probability of a Product Being Defective $(x)$:$x\%$ over the entire production, where $x$ is equal to the last two digits of your student number plus $1$ divided by $100.$ For example, if your student number is $AD202341,$ then $x=\frac{41+1}{100}=0\mathrm{.}42.($Student number for example ends in $12)$

Sensitivity of the Testing Process $($True Positive Rate$)$: $93\%($Correct positive result when a product is defective$).$

Specificity of the Testing Process $($True Negative Rate$)$: $94\%($Correct negative result when a product is not defective$).$

In other cases, the test returns the opposite result.

a$)$ Summarize the above problem description using probabilities.

b$)$ Suppose we now observe a new product from the manufacturing plant for which the test returns a positive result. What is the probability of the product being defective$/$not defective?