Week 6: Self-Guided Problems

When you come to class, be ready to discuss these questions, and bring any additional questions you still have after this exercise!

For the first week only, this Self-Guided exercise is due by Friday. All other Self-Guided Problem Sets will be due before class starts.

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.

2. Compile into a PDF. This can be done easily through an Overleaf account, otherwise you will need to install LaTeX.
3. Submit the pdf to Gradescope via the appropriate submission link for the course.
4. 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).

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.

Self-Guided Problems cannot be submitted late for any reason

This is due to the fact we discuss them in the class. Be sure to stay on top of these Self-Guided problems, and remember it is better to turn in an incomplete set than an empty one.

ChatGPT/AI

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

NEW: Forming Groups

You may solve Self-Guided Problem sets in groups of up to 4 people.

Rules for groups:

• You must list everyone in your group on the assignment. Leaving off a name counts as academic dishonesty.
• Working with another group/working with more than 4 people counts as academic dishonesty.
• Homework assignments, exams, and projects are still individual assignments, and any group work on these counts as academic dishonesty.
• You must have a table at the start of your problem sets with approximate "effort" percentages. For example, if I work with David, and I think we both contributed equally to the problem set, I would insert a table:

If your effort percentage is very low or consistently low, your score may be changed to reflect your effort.

If a group has problems deciding the amount of effort each member should recieve, the group will not be permitted to work together again for an assignment.

• Every member of the group must understand the solution to every problem in the Self-Guided Problem set.
• All other guidelines still apply to Self-Guided Problem Sets (no use of AI tools, no plagiarism, must be typeset in LaTeX, etc.)

NEW: Assignment Template

Example templates for Self-Guided Problem sets and Homework assignments are available in the shared Overleaf Document, under Templates.

Do not under any circumstances modify the templates in the Shared Overleaf.

There is a list at the bottom of both assignments, meant for citation. If a resource was consulted, it should be linked, and the problem(s) it has been used for should be specified.

As a note: Copying answers word-for-word from any sources is considered plagiarism. This may result in a zero on the problem or a zero on the assignment depending on severity. Any use of ChatGPT or other AI tools will result in a zero on the assignment.

As a general guideline: if you could not do the problem yourself without looking at the resource, you should not be using this resource and should find another way to study.

I would also generally advise against using resources such as Quora, Reddit, etc where people crowdsource answers. People can say many things on the internet, including things that are untrue or misleading. The best resources will always be the readings on Anchor, the textbook, live classes, Discord, and my office hours.

Problems

1. Prove one of the Identities based in Pascal’s Triangle using:

a. Blockwalking

b. A Combinatorial Proof

c. Algebra

Do not pick an identity that was proven on Anchor.

d. Which method was easiest for this identity? What do you think made it easiest?

2. Create a 6-sided die where

a. The chances of rolling a one are 2/3.

b. The chances of rolling an even number are 100%.

c. The probability of rolling a 3 is 0%.

d. The probability of the sum of two of your die is 8 is 1/6.

3. Determine if the following events are independent or dependent. Note that you do not have to solve these problems, just determine if they are independent or dependent.

a. Drawing a red card, then drawing a card labeled k.

b. Drawing a red card, then drawing another red card.

c. Drawing a red card, then rolling a 6 on a standard, fair, 6-sided die.

d. Rolling a 6 on a standard, fair, 6-sided die, then rolling a 1 on the die.

4. Find the probability of the following events. Assume all dice are standard, fair 6-sided dice.

a. Rolling 2 dice and they are both ones.

b. Rolling a one, then rolling another one.

c. Rolling an even number, then an odd number.

d. Rolling an odd number, then an even number.

e. Rolling an even number, then an even number.

f. Rolling an even number, then a different even number.

5. Imagine there's a test for a disease that only 2% of people have. The test has a 99% sensitivity rate, and a 95% 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?

6. Find the expected value of

a. Rolling a standard, fair 8-sided die.

b. The sum of two rolls of a standard, fair, 8-sided die.

c. The sum of four rolls of a standard, fair 8-sided die.