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Most likely coin-tossing sequence. In Parade Magazine鈥檚 (November 26, 2000) column 鈥淎sk Marilyn,鈥� the following question was posed: 鈥淚 have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine.鈥�

(1) H HHHHHHHHH

(2) H H T T H T T H HH

(3) T TTTTTTTTT

1. Demonstrate that prior to actually tossing the coins, thethree sequences are equally likely to occur.
2. Find the probability that the 10 coin tosses result in all heads or all tails.
3. Find the probability that the 10 coin tosses result in a mix of heads and tails.
4. Marilyn鈥檚 answer to the question posed was 鈥淭hough the chances of the three specific sequences occurring randomly are equal . . . it鈥檚 reasonable for us to choose sequence (2) as the most likely genuine result.鈥� If you know that only one of the three sequences actually occurred, explain why Marilyn鈥檚 answer is correct. [Hint: Compare the probabilities in parts b and c.]

Simulate the experiment described in Exercise 3.7 using any five identically shaped objects, two of which are one colour and the three another colour. Mix the objects, draw two, record the results, and then replace the objects. Repeat the experiment a large number of times (at least 100). Calculate the proportion of time events A, B, and C occur. How do these proportions compare with the probabilities you calculated in Exercise 3.7? Should these proportions equal the probabilities? Explain.

Confidence of feedback information for improving quality. In the semiconductor manufacturing industry, a key to improved quality is having confidence in the feedback generated by production equipment. A study of the confidence level of feedback information was published in Engineering Applications of Artificial Intelligence(Vol. 26, 2013). At any point in time during the production process, a report can be generated. The report is classified as either 鈥淥K鈥� or 鈥渘ot OK.鈥� Let Arepresent the event that an 鈥淥K鈥� report is generated in any time period (t).Let Brepresent the event that an 鈥淥K鈥� report is generated in the next time period. Consider the following probabilities:

$\mathbf{P}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{8}$,$\mathbf{P}\left(\frac{B}{A}\right)\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}$, and$\mathbf{P}\left(\frac{B}{{\mathbf{A}}^C}\right)\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}$.

a. Express the event B|Ain the words of the problem.

b. Express the event B|${\mathbf{A}}^C$in the words of the problem.

c. Find$\mathbf{P}\mathbf{(}{\mathbf{A}}^C\mathbf{)}$.

d. Find$\mathbf{P}\mathbf{(}\mathbf{A}鈭�\mathbf{B}\mathbf{)}$.

e. Find$\mathbf{P}\mathbf{(}{\mathbf{A}}^C鈭�\mathbf{B}\mathbf{)}$.

f. Use the probabilities, parts d and e, to find P(B).

g. Use Bayes鈥� Rule to find P(A|B), i.e., the probability that an 鈥淥K鈥� report was generated in one time period(t), given that an 鈥淥K鈥� report is generated in the next time period$\left(t+1\right)$.

Exit poll candidates and voters.In an exit poll, 45% of voters said that the main issue affecting their choice of candidates was the economy, 35% said national security, and the remaining 20% were not sure. Suppose we select one of the voters who participated in the exit poll at random and ask for the main issue affecting his or her choice of candidates.

1. List the sample points for this experiment.
2. Assign reasonable probabilities to the sample points.
3. What is the probability that the main issue affecting randomly selected voters鈥� choice was either the economy or national security?

From a production batch with 16 items, 8 items are randomly selected for quality assurance. In how many different ways can the sample be drawn? Suggest an estimate before computing the exact number.

The Venn diagram below illustrates a sample space containingsix sample points and three events, A, B, and C.The probabilities of the sample points are $\mathbf{}\mathbf{P}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{.}\mathbf{3}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{}\mathbf{.}\mathbf{2}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{3}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{.}\mathbf{1}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{.}\mathbf{1}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{5}\mathbf{\right)}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{.}\mathbf{1}$,and$\mathit{P}\mathbf{\left(}\mathbf{6}\mathbf{\right)}\mathbf{=}\mathbf{.}\mathbf{2}$.

a. Find$\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{鈭�}\mathbf{B}\mathbf{)}\mathbf{,}\mathbf{P}\mathbf{(}\mathbf{B}\mathbf{鈭�}\mathbf{C}\mathbf{)}\mathbf{,}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{鈭�}\mathbf{C}\mathbf{)}\mathbf{,}\mathbf{P}\mathbf{(}\mathbf{A}\mathbf{鈭�}\mathbf{B}\mathbf{鈭�}\mathbf{C}\mathbf{)}\mathbf{,}\mathbf{P}\mathbf{\left(}{\mathbf{B}}^{\mathbf{c}}\mathbf{\right)}\mathbf{,}\mathbf{P}\mathbf{(}{\mathbf{A}}^{\mathbf{c}}\mathbf{鈭�}\mathbf{B}\mathbf{)}$.

b. Are A and B independent? Mutually exclusive? Why?

c. Are B and C independent? Mutually exclusive? Why?

Management system failures. Refer to the Process Safety Progress (December 2004) study of 83 industrial accidents caused by management system failures, Exercise 2.150(p. 142). A summary of the root causes of these 83 incidents is reproduced in the following table. One of the 83 incidents is randomly selected and the root cause is determined.

a. List the sample points for this problem and assign reasonable probabilities to them.

b. Find and interpret the probability that an industrial accident is caused by faulty engineering and design.

c. Find and interpret the probability that an industrial accident is caused by something other than faulty procedures and practices.

Workers鈥� unscheduled absence survey. Each year CCH, Inc., a firm that provides human resources and employment law information, conducts a survey on absenteeism in the workplace. The latest CCH Unscheduled Absence Surveyfound that of all unscheduled work absences, 34% are due to 鈥減ersonal illness,鈥� 22% for 鈥渇amily issues,鈥� 18% for 鈥減ersonal needs,鈥� 13% for 鈥渆ntitlement mentality,鈥� and 13% due to 鈥渟tress.鈥� Consider a randomly selected employee who has an unscheduled work absence.

a. List the sample points for this experiment.

b. Assign reasonable probabilities to the sample points.

c. What is the probability that the absence is due to something other than 鈥減ersonal illness鈥�?

Ownership of small businesses. According to the Journal of Business Venturing (Vol. 17, 2002), 27% of all small businesses owned by non-Hispanic whites nationwide are women-owned firms. If we select, at random, a small business owned by a non-Hispanic white, what is the probability that it is a male-owned firm?

Condition of public school facilities. The National Center for Education Statistics (NCES) conducted a survey on the condition of America鈥檚 public school facilities. The survey revealed the following information. The probability that a public school building has inadequate plumbing is .25. Of the buildings with inadequate plumbing, the probability that the school has plans for repairing the building is .38. Find the probability that a public school building has inadequate plumbing and will be repaired.