Your company is considering offering $400$ employees the opportunity to transfer to its new headquarters in Ottawa and, as personnel manager, you decide that it would be fairest if the transfer offers are decided by means of a lottery. Assuming that your company currently employs $300$ managers, $500$ factory workers, and $900$ miscellaneous staff, find the following probabilities, leaving the answers as formulas.

$($a$)$ All the managers will be offered the opportunity.

C

Incorrect: Your answer is incorrect.

$,$

Incorrect: Your answer is incorrect.

C

$,$

$($b$)$ You will be offered the opportunity.

C

$,$

C

$,$