ECE 108 - Assignment 10
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ECE 108 - Assignment 10
Submit to Crowdmark by 9:00pm EST on Friday, April 7.
Instructions:
1. Answer each question in the space provided or on a separate piece of paper. You may also use typetting software (e.g., Word, TeX) or a writing app (e.g., Notability).
2. Scan or photograph your answers.
3. Upload and submit your answers by following the instructions provided in an e-mail sent from Crowd- mark to your uWaterloo e-mail address.
4. Close the Crowdmark browser window. Follow your personalized Crowdmark Assignment link again to carefully view your submission and ensure it will be accepted for credit. Any pages that are uploaded improperly (sideways, upside down, too dark/light, text cut off, out of order, in the wrong location, etc.) will be given a score of zero. Otherwise, they will be graded as either complete or incomplete.
Unless otherwise stated, your final answer can be left in terms of P(n,m), (m(n)) and ((m(n))).
You must show your work/reasoning for credit. A solution with no justification will receive no credit. Questions:
1. (3 points) Prove or disprove: Given any set S, there exists a function f : P(S) → R that satisfies the three Kolmogorov axioms.
2. (6 points) The symmetric difference of sets A and B, denoted by A∆B, is the set (A−B)n(B−A). Let S be a sample space, let Pr be a PDF on S and let A,B ∈ P(S).
(a) Prove that if A and B are mutually exclusive then Pr{A∆B} = Pr{A n B}. (b) Prove or disprove the converse of the implication in part (a).
In this question, you are allowed to make informal arguments about sets using Venn diagrams.
3. (5 points) Marmie is excited that the critically acclaimed game “The Meow of Us Part 1”, which was a MeowStationTM exclusive, is now being ported to PC. Because of how important the “The Meow of us” IP is to Meowny, the company that makes it, Marmie believes that there is a 90% chance that the port will be good.
Unfortunately, Marmie just found out that the Iron Kitten company is making the port. The Iron Kitten is responsible for making 1% of all MeowStation to PC ports and it is known that Iron Kitten has made 3 of the 30 good MeowStation to PC ports.
What is Marmies’ updated probability that the “The Meow of Us Part 1” PC port is good given that that Iron Kitten is making the port?
4. (3 points) Suppose we roll eight fair D6 and sum the pips. What is the probability that the sum is
19 given that we know that exactly 6 of the 8 dice rolled a 2.
5. (2 points) Prove or disprove: If we roll 2d6 then the event that the first die is even is independent from the event that the sum is 11.
6. (5 points) You are using an AI to determine if an image contains a cat or not. Suppose the probability that a random picture contains a cat is known to be 6%. Before using the AI to classify pictures, you test the AI on a data set containing 5000 images that have been previously been classified by an expert cat photo identifier. Exactly 1000 of your test images do not contain a cat and the remaining images do in fact depict a cat. Your AI correctly identifies 800 of the images that do not contain a cat and 3600 of the images that do contain a cat.
Using the information above, compute the probability that a random picture we found from the internet is a cat given that the AI model classifies it as containing a cat.
You can leave your final answer as a probability with four digits.
7. (3 points) Suppose there is a game where you spend $1.50 to flip N fair coins. If this results in exactly i heads you get i dollars.
(a) Find an expression the amount of money you expect to receive from playing this game. Your
answer can be written in summation notation.
(b) Find the smallest value of N where you expect to make a profit by playing the game.
2023-04-04