A machine can develop two kinds of problems: P$1$ and P$2.$ Depending upon the kind of problem, it can start making a knocking noise $($KN$),$ or it can start going low on power $($LP$),$ or it can show both symptoms. Both symptoms manifest independently of the other. Here is the probability table that shows the likelihood of seeing these symptoms:

Problem

KN

LP

No Problem

$0\mathrm{.}01$

$0\mathrm{.}05$

P$1$

$0\mathrm{.}3$

$0\mathrm{.}6$

P$2$

$0\mathrm{.}4$

$0\mathrm{.}8$

P$1$ and P$2$

$0\mathrm{.}8$

$0\mathrm{.}95$

Further, when a machine does not have a problem, it has a $10\%$ probability of developing each of those problems, but cannot develop both problems at once. When it has one of the problems, it has a $20\%$ probability of developing the other problem as well. Once the machine develops a problem, the problem never goes away.

a$)$ Draw and explain an HMM to model this problem.

b$)$ You make the following observations:

None, LP$,$ LP$,$ None, LP$,$ LP and KN$,$ LP and KN$.$

What is the best explanation for the problems that the machine is developing?

PLease refer to the attached picture for the question