# BUA Probability and Simulatio

Q1.A production manager believes that the machine used to produce items in their factory is not consistent. In fact, the manager believes that, on a given day,the machine produces defective items with probabilities 0.1,0.2,0.3,0.4, 0.5, 0.6, 0.7, 0.8, and0.9(say p). Furthermore, the managerβs prior belief indicates that each probability (p) is equally likely.During the last 30 days, the machine produced 7 defective items.Assuming that each item is produced independently from each other, obtain the probability that p=0.1, 0.2, …, 0.9 (individually)given that themachine produced 7 defective items in the last 30 days.Which posterior probability for p has the highest support? What does this mean?

Q2.Suppose that an online retailer has 350 items in store (on a given day), of which 250 are books and 100are board games. If 30 items are purchased (randomly) from the store, what is the probability that the order will contain 18 books?

Q3.A programmer is about to attempt to compile a series of 11 similar programs. Let π΄πbe the event that the ith program compiles successfully for i=1,…,11. When the programming task is easy, the programmer expects that 80 percent of programs should compile. When the programming task is difficult, the programmer expects that only 40 percent of the programs will compile. Let B the event that the programming task was easy. The programmer believes that the events π΄1,π΄11are conditionally independent givenπ΅and π΅π.

a) Compute the probability that exactly 8 out of 11 programs will compile given that the task was easy.

b) Compute the probability that exactly 8 out of 11 programs will compile given that the task was difficult.

Q4.* A study by Forbes indicated that the five most common words appearing in spam emails are shipping!, today!, here!, available! andfingertips!. Suppose that for one email account, 1 in every 10 messages is spam and the proportionsof spam messages that havethe five most common words in a spam email are given as follows: shipping!β0.051 today!β0.045here!β0.034 available! β0.014 fingertips!β0.014 Also suppose that the proportions of ham messages (emails that are not spam) that have these words are shipping!β0.0015 today!β0.0022here!β0.0022 available! β0.0041 fingertips!β0.0011

a)If a message includes the word shipping!,what is the probability that the message is spam? and what is the probability that it is ham?

b)If a message includes the word today!,what is the probability that the message is spam? and what is the probability that it is ham?

c)If a message includes the word here!,what is the probability that the message is spam? and what is the probability that it is ham?

d)If a message includes the word available!,what is the probability that the message is spam? and what is the probability that it is ham?

e)If a message includes the word fingertips!,what is the probability that the message is spam? and what is the probability that it is ham?

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