Problem $3\mathrm{.}8($Deterministic dynamic programming$)$: Michael Jackson needs to pass all the three subjects he is taking this semester. He is now enrolled in Marketing, Business Modelling, and Ballroom Dancing. Michael's busy schedule of extracurricular activities $($i$.$e$.,$ drinking beers and chasing girls$)$ allows him to spend only $4$ hours per week on studying. Michael's probability of passing each subject depends on the number of hours he spends studying for the course as follows: $(4$ points$)$

$\backslash$table$[[\backslash$table$[[$Hours study$],[$per week$]],$Probability of passing a course$],[0,0\mathrm{.}75,0\mathrm{.}70,0\mathrm{.}80],[1,0\mathrm{.}80,0\mathrm{.}75,0\mathrm{.}90],[2,0\mathrm{.}85,0\mathrm{.}80,0\mathrm{.}92],[3,0\mathrm{.}86,0\mathrm{.}82,0\mathrm{.}95],[4,0\mathrm{.}90,0\mathrm{.}85,0\mathrm{.}99]]$

$($a$)$ This problem can be formulated as a dynamic programming problem as shown below:

Stage, $i$ :

State, $s_i$ :

Decision, $d_i$ :

$i$ th subject, where $i=1,2,$ and $3.$

Hours available at the current stage.

Hours spent on subject $i.$

Let $p_i(d_i)$ denote the probability of passing a subject $i$ if he spends $d_i$ hours on it$,$ and let $f_i(d_i,s_i)$ be the maximum probability obtained at stage i when, among $s_i$ hours still available, he spends $d_i$ hours on subject $i.$ Find the optimality equation.

$($b$)$ Using a spreadsheet software, set up the optimality table for the dynamic programming problem and find the optimal decision strategy for Michael. What is the maximum probability that he will pass all the three subjects?