Bayes' Theorem/Examples

Examples of Use of Bayes' Theorem

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 $2$?

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 $X$?

Let $C_1$ be a a double-headed coin.

Let $C_2$ be a fair coin.

Let $A_1$ denote the event of tossing $C_1$.

Let $A_2$ denote the event of tossing $C_2$.

Suppose one of $C_1$ and $C_2$ is chosen at random with $\map \Pr {A_1} = \dfrac 1 2$ and $\map \Pr {A_2} = \dfrac 1 2$.

Let $B$ be the event of landing heads.

What is the probability that it was $C_1$ that was tossed?