2S2122 Activity 1.1

Instructions

• Answer the following problems individually and truthfully.
• Be sure to show your solutions and please box your final answers.
• Please write your complete name, student number, and section on the upper left corner of your answer sheet. No name, student number, and section, no grade.
• Save your answers in pdf file type with the filename format "section_lastname_firstname_studentnumber.pdf" all in small caps. For example: "abc_wayne_bruce_201101474.pdf".
• Submit your files in the respective submission bin in UVLE. Be sure to submit in the correct class!
• Have fun doing these exercises :) even though it may seem boring.
• You only have two weeks to work on the activities. We will post the hard deadline on UVLE.

Bonus: If you want to submit an elegant work, you may want to try Overleaf. It's a document processing site where you encode your work using Latek. It's about time you learn this because you will use this when you graduate.

Grading Rubrics

• If you have a good solution and a correct answer, you get full points.
• If you have a good solution but the answer is not boxed (or highlighted), you get a 5% deduction of the total points for that problem.
• If you have a good solution but the answer is wrong, you get a 20% deduction of the total points for that problem.
• If your solution is somewhat OK but incomplete. You only get 40% of the total problem.
• If you have a bad solution but with a correct answer. That sounds suspicious. You get 0% for that problem. A bad solution may be:
  • You just wrote the given.
  • You just dumped equations but no explanation to where they are used.
  • You attempted to put a messy flow to distract us. That's definitely bad.
• No attempt at all means no points at all.

Problem 1 (2 pts.)

Three fair coins are tossed successively.

(a) Write down the sample space. (0.1 pts)

(b) What are the events for when the first coin is a head? Let this be event $E$. (0.15 pts)

(c) What are the events for when the second coin is a tail? Let this be event $F$. (0.15 pts)

(d) Calculate $P(E\cup F)$. (0.4 pts)

(e) Calculate $P(E\cap F)$. (0.4 pts)

(f) Calculate $P({\bar {E}})$. (0.4 pts)

(g) Calculate $P({\bar {F}})$. (0.4 pts)

Problem 2 (2 pts.)

Two events $A$ and $B$ are such that $P(A)=0.45$, $P(B)=0.22$, and $P(A\cup B)=0.53$. Find $P({\bar {A}}\cup {\bar {B}})$. (2 pts)

Problem 3 (2 pts.)

In a factory producing IC chips, the total quantity of defective items found in a given week is $14\%$. It is suspected that the majority of these come from two machines, $X$ and $Y$. An inspection shows that $8\%$ of the output from $X$ and $4\%$ of the output from $Y$ is defective. Furthermore, $11\%$ of the overall output came from $X$ and $23\%$ from $Y$. An IC chip is chosen at random and found out to be defective. What is the probability that it came from either $X$ or $Y$? (*Hint*: Read very carefully.) (2 pts.)

Problem 4 (2 pts.)

In eight tosses of a fair coin, find the probability that heads will come up:

(a) Exactly three times. (0.5 pts.)

(b) At least three times. (0.75 pts.)

(c) At most three times. (0.75 pts.)

Problem 5 (2 pts.)

Prove the following:

(a) Given some random variable $X$ and some constant $c$, show that ${\textrm {Var}}(X+c)={\textrm {Var}}(X)$. (1 pts.)

(b) Given some two independent random variables $X$ and $Y$, show that ${\textrm {Var}}(X-Y)={\textrm {Var}}(X+Y)$. (1 pts.)