Two students are randomly selected from a class of $12$ girls and $18$ boys.

Find the expected number of girls and the expected number of boys.

Two students are randomly selected from a class of $12$ girls and $18$ boys.

Write the ratio of the expected number of girls to the expected number of boys in simplified form. What do you notice about this ratio?

Two students are randomly selected from a class of $12$ girls and $18$ boys.

Calculate the variance of the number of girls selected.

A sewing basket contains eight reels of cotton: four are green, three are red and one is yellow. Three reels of cotton are randomly selected from the basket.

Show that the expected number of yellow cotton reels is $0.375.$

A sewing basket contains eight reels of cotton: four are green, three are red and one is yellow. Three reels of cotton are randomly selected from the basket.
Find the expected number of red cotton reels.

A sewing basket contains eight reels of cotton: four are green, three are red and one is yellow. Three reels of cotton are randomly selected from the basket.
Hence, state the expected number of green cotton reels.

A company offers a $\$1000$ cash loan to anyone earning a monthly salary of at least $\$2000.$ To secure the loan, the borrower signs a contract with a promise to repay the $\$1000$ plus a fixed fee before $3$ months have elapsed. Failure to do this gives the company a legal right to take $\$1540$ from the borrower's next salary before returning any amount that has been repaid.

From past experience, the company predicts that $70\%$ of borrowers succeed in repaying the loan plus the fixed fee before $3$ months have elapsed.

Calculate the fixed fee that ensures the company an expected $40\%$ profit from each $\$1000$ loan.

A company offers a $\$1000$ cash loan to anyone earning a monthly salary of at least $\$2000.$ To secure the loan, the borrower signs a contract with a promise to repay the $\$1000$ plus a fixed fee before $3$ months have elapsed. Failure to do this gives the company a legal right to take $\$1540$ from the borrower's next salary before returning any amount that has been repaid.

From past experience, the company predicts that $70\%$ of borrowers succeed in repaying the loan plus the fixed fee before $3$ months have elapsed.

Assuming that the company charges the fee found in part a, how would it be possible, without changing the loan conditions, for the company's expected profit from each $\$1000$ loan to be greater than $40\%$ ?

When a scout group of $8$ juniors and $12$ seniors meets on a Monday evening, one scout is randomly selected to hoist a flag. Let the variable $X$ represent the number of juniors selected over $n$ consecutive Monday evenings.

By drawing up the probability distribution table for $X,$ or otherwise, show that $E\left(X\right)=1.2$ when $n=3.$