In a remote work setup, you rely on two different internet connections to ensure uninterrupted work: a primary fiber optic connection with a $0.003$ probability of failure and a backup satellite connection with a $0.020$ probability of failure. What is the probability that both internet connections fail? What is the probability that both do not fail?

During a survey on a new diet plan, $120$ individuals reported weight loss while $80$ reported no weight loss. It was later found that $20$ of the weight loss reports were inaccurate (false positives), and $2$ of the no-weight-loss reports were incorrect (false negatives). How many individuals participated in the survey?

A rare genetic disorder affects $1$ out of every $500$ individuals. A screening test for the disorder returns a positive result $90\%$ of the time when a person actually has the disorder, and $4\%$ of the time when a person does not have the disorder. If a person tests negative, what is the probability that they do not have the disorder? Use Bayes' Theorem.

A popular word game presents players with a scrambled set of letters: V, E, S, E, L, L.

i. How many ways can these letters be arranged?

ii. If one arrangement is randomly selected, determine the probability that it forms a correctly spelled word.

If you have 2 types of bread, 3 types of meat, and 4 types of cheese, how many different sandwiches can you make?

A security badge code is made up of $5$ characters. Each character can be any uppercase letter from A to Z or any digit from $0$ to $9$, except that the first character must be a letter and the last character must be an even digit. What is the probability of guessing the correct badge code on the first attempt?

A library has $10$ books available for donation. The library staff wants to select $3$ different books from this collection to display on a special shelf. Additionally, they need to choose one of the displayed books to label as the 'Book of the Month.' If the books are selected and labeled at random, what is the probability of selecting the $3$ specific books the staff had in mind for the display, as well as one exact book for the 'Book of the Month' label? (Note: That the $3$ specific books the staff had in mind for the display are predefined and are known in advance.)

A committee of $4$ is to be formed from $5$ men and $7$ women. What is the probability that the committee will have exactly $2$ men and $2$ women?

The following table shows the number of secondary school students (in thousands) enrolled in Illinois and Florida by grade. If a student is chosen at random from all those listed, what is the probability that the student is enrolled in Florida, given that the student is in the $11$^th grade?