Q$4.$ Suppose there are two types individuals are interested in a health insurance: High risk $\backslash ((\backslash$mathrm$\{$H$\})\backslash )$ and Low risk $\backslash ((\backslash$mathrm$\{$L$\})\backslash ).$ The chance an H person gets sick is $\backslash (5\backslash \%\backslash )$ and the cost of an H person being sick is $\backslash (\backslash$$ $10,000\backslash ).$ The chance an L person gets sick is $\backslash (1\backslash \%\backslash )$ and the cost of an L person being sick is $\backslash (\backslash$$ $1000\backslash ).$ There are $100$ H individuals and $900$ L individuals in the population. Find:

$($i$)$ The expected cost of insuring a random person from the population before any selection effects occur.

$($ii$)$ The fraction of sick people who are uninsured after accounting for selection effects.

$($iii$)$ The fraction of medical expenses that are not covered by insurance after accounting for selection.