Lending Club is the largest peer$-$to$-$peer $($P$2$P$)$ lending company in the United States. The company provides a fully online lending platform directly connecting borrowers and investors. By cutting out banks from the process, borrowers typically get lower rates on loans. In applying for a loan, borrowers state the purpose of their loan request. The most common purposes are paying off credit cards $($CC$),$ debt consolidation $($DC$),$ and funding a home improvement project $($HI$).$

Among the information gathered is the borrower's dwelling status: home paid in full $($Own$),$ has a mortgage $($Mortgage$),$ and renting an apartment $($Rent$).$

Following are data on nearly $120,000$ approved loans for the fourth quarter of $2020$ organized as a two$-$way table of counts:

$\backslash$table$[[,$Dwelling status$],[$Purpose$,$Own,Mortgage,Rent,Total$],[$Credit card $($CC$),3265,11,396,10,645,25,306],[$Debt consolidation $($DC$),7245,30,476,24,731,62,452],[$Home improvement $($HI$),1432,6430,816,8678],[$Other$,3358,8758,10,094,22,210],[$Total$,15,300,57,060,46,286,118,646]]$

a$.$ Randomly choose a borrower from the $118,646$ borrowers. What is the probability that a borrower looks for a loan for a home improvement project?

b$.$ Now let's consider the probability that a randomly chosen borrower has a mortgage and looks for a loan for a home improvement project.

$(10$p$)$

The S&P $500$ index has a probability $0\mathrm{.}56$ of increasing in any week. Moreover, the change in the index in any given week is not influenced by whether it rise or fell in earlier weeks. Let $x$ be the number of weeks among the next five weeks in which the index rises.

a$.x$ has a binomial distribution. What are $n$ and $p?$

b$.$ What are the possible values that $x$ can take?

c$.$ Use the binomial formula to find the probability of $x=5.$

d$.$ What are the mean and standard deviation of this distribution?