Bayes' Theorem/Examples

Examples of Use of Bayes' Theorem

Arbitrary Example $1$

Let box $1$ contain $10$ good screws and $2$ unslotted screws.

Let box $2$ contain $8$ good screws and $4$ unslotted screws.

Let a box be selected at random.

Let a screw chosen from that box prove to be unslotted.

What is the probability that it came from box $2$?

Arbitrary Example $2$

Suppose that, in a population, $6$ out of every $1000$ people has an illness $X$.

It is known that:

if a person has $X$, there is a $92 \%$ probability that a blood test will be positive for $X$

if a person does not have $X$, there is a $0 \cdotp 5 \%$ probability that a blood test will be positive for $X$.

Let a person selected at random test positive for $X$.

What is the probability that this person actually has $X$?