**Instructor: Ali Nowroozi**

**Fall 2024**

__Homework Assignment #2__

__DUE: 10/06/2024 11.59 PM__

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**(30 Points)**

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**Instructions:**

- Answer all the questions in this file. Do not delete the questions. Type in your answers below the question, using the sample structure (if provided)
- We expect precise and succinct answers. Add citations, references, and screenshots wherevernecessary.
- Follow the deadline for submission. Late submissions will not be accepted and will receive azero.
- Name the file as the
and submit it__last four digits of your USC ID____as a____PDF__^{(1)}. (For Example, if myUSC ID is 123456789, the file name becomes: 6789.pdf).__File__.__Do not put your name anywhere on this document or on the file name or you will lose 2 points__ - Not following the instructions will result in penalties.

*To convert this document to a PDF file, go to “file”, “save As”, choose “PDF(*.PDF)” and the file will be saved at the same location as your WORD file.*

- Which pair of possibilities is “Mutually Exclusive”, but
__not__“Collectively Exhaustive”? Draw the correct MECE diagram () from the attached model.__for the answer only__- The temperature outside is: under 10
^{o}C , Over 20^{o}C - The temperature outside is: under 20
^{o}C , Over 10^{o}C - The temperature outside is: under 20
^{o}C , Over 20^{o}C - The temperature outside is: under 10
^{o}C , Under 20^{o}C

- The temperature outside is: under 10

*Tip: Refer to Lecture 2-Slide 16, also see the attached models*

- Sara has three coins which you believe to be fair, and are labeled #1, #2, and #3. Suppose Sara flips all three coins and tells you that
2 of them landed heads. Given that you believe each coin’s probability of landing heads is ½ and it is irrelevant to how the other coins land, use an event tree model with all consequences listed to show that the probability that coin #1 landed on Head is ¾.__at least__

*Tip: flipping 3 coins is the same as flipping the same coin 3 times, for the purpose of this experiment. So, you can use our model in slide 22 of lecture 3. No equation is necessary, just count the outcomes with the desired features.*

- What is the probability of drawing 2 Aces from 52 cards?

*Tip #1: The probability of one item in a group of N items is 1/N*

*Tip #2: This is the probability that the first card is an ACE (say event A), AND the second card is an ACE (say event B) *

*è*

*, but A & B are independent so P(B|A) = P(B)*- Use Bayesian law to show that
that a driver who is over age 35 wears a seat belt regularly in the following scenario:__there is a 64% chance__

*ABC Insurance Co. estimates that 80% of drivers wear seat belts regularly. They also estimate that 50% of drivers are over age 35. A study showed that 40% of those drivers who wear seat belts regularly are over age 35. *

*Tip: Event A = Wear Seatbelt, Event B = Over age 35, find P(A|B)*

- See attached a fault tree example with calculations. Identify a
problem on your own, develop a__different__fault tree (with less than 10 nodes, like the attached example) and assign some probabilities to each node and calculate the overall probability of the event (e.g. a house getting on fire in one year in UK). Now investigate the actual probability data on the internet (e.g. this is what I found for my example). If you see a big difference, revisit your assumptions on the individual probabilities, adjust them, and explain your adjustments briefly.__simple__

*Tip #1: If you model your problem in Excel, the adjustments will be very easy and quick *

*Tip #2: In the example, P(A **ꓴ B) is calculated differently. Please ignore this method and use the simple formula discussed in the class: P(A **ꓴ B) = P(A) + P(B) , A and B are independent*

*Source: **https://careerinconsulting.com/wp-content/uploads/2021/03/Capture-MECE-Diagram-1024×577.png*

*Source: **https://youtu.be/Eq8-m6Faobo?si=MGm9iGOLQJb-FNV8*