Homework Set 6

Submitting Your Work

Your work must be submitted to Anchor for degree credit and to Gradescope for grading.

For any work completed outside of GitHub or Gradescope:

  1. Typeset your work in LaTeX.
  • You should use \begin{enumerate} to create a numbered list, and your solutions to each problem should match with the corresponding number on the assignment.
  • For example, if an assignment has 10 numbered problems, your enumerate should have 10 \item commands, with your solution(s) to problem 1 under the first \item, problem 2 under the second and so on.
  • If you skip a problem, just leave the \item blank, otherwise the numbers of your solutions will not match the assignment document. Here is an example Overleaf document showcasing this format.
  1. Compile into a PDF. This can be done easily through an Overleaf account, otherwise you will need to install LaTeX.
  2. Submit the pdf to Gradescope via the appropriate submission link for the course.
  3. Upload the pdf to Anchor using the form below.

Note: Anchor submissions can occur at any time during the term, but it is critical that you upload all of your work to Anchor before the last day of the term. Gradescope submissions must be submitted before the deadline (or the late deadline, if applicable).

ChatGPT/AI

Use of ChatGPT/AI is forbidden for all assignments in this course.

It is required that all assignments are submitted as a PDF generated from a LaTeX document.

Assignments submitted in any other form will earn zero credit.

  1. Using a combinatorial proof, show that the sum of the $n$th row in Pascal's triangle is $2^n$

  2. Using a combinatorial proof, show that ${4n \choose 2} = 4 \cdot {n \choose 2} + 6n^2$

Hint: $6 = {4 \choose 2}$ and $n$ can also be written as ${n \choose 1}$

  1. Find the probabilities of the following events:

a. A randomly chosen integer number between 1 and 100 contains a 2.

b. Rolling a sum of 3 on two standard, fair, 6-sided die.

c. Rolling a sum of 3 on three standard, fair 6-sided die.

d. Drawing all 4 letters from our Kibo deck in a hand of 5 cards.

e. A randomly chosen real number chosen between 1 and 10 falls in the range from 2 to 4.

f. A point randomly selected in a square of side length 1 also appears in a circle centered on the square with radius $\frac{1}{2}$.

  1. Imagine there's a test for a disease that only 0.5% of people have. The test has a 99.9% sensitivity rate, and a 99% specificity rate.

a. If a person has a positive result, what is the probability they have the disease?

b. If a person has a negative result, what is the probability they do not have the disease?

  1. Find the expected value of the following events

a. The sum of numbers on 5 cards drawn from our Kibo deck.

b. The sum of numbers on 10 cards drawn from our Kibo deck.

c. The outcome of 4 coin flips where heads add 1 and tails subtract 1.

d. The multiplication of two rolls of a standard, fair 6-sided die.

  1. Draw or graph a probability mass function for the following:

a. The sum of two dice rolls.

b. The results of a single coin flip.

c. The value of a random integer number between 1 and 10.