Bayes' Theorem/Examples/Arbitrary Example 1

Example 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 $2$?

Let $A$ be the event: An unslotted screw is selected.

Let $B_1$ be the event: Screw is selected from box $1$.

Let $B_2$ be the event: Screw is selected from box $2$.

We have that:

$\ds \map \Pr {B_1} = \map \Pr {B_2}$

$=$

$\ds \dfrac 1 2$

$\ds \condprob A {B_1}$

$=$

$\ds \dfrac 1 6$

$\ds \condprob A {B_2}$

$=$

$\ds \dfrac 1 3$

$\ds \leadsto \ \ $

$\ds \condprob {B_2} A$

$=$

$\ds \dfrac {\map \Pr {B_2} \condprob A {B_2} } {\map \Pr {B_1} \condprob A {B_1} + \map \Pr {B_2} \condprob A {B_2} }$

$\ds $

$=$

$\ds \dfrac {\frac 1 2 \times \frac 1 3} {\frac 1 2 \times \frac 1 6 + \frac 1 2 \times \frac 1 3}$

plugging in the numbers

$\ds $

$=$

$\ds \dfrac 2 3$

calculation

Hence the probability is $\dfrac 2 3$ that the unslotted screw came from box $2$.

$\blacksquare$