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Chapter 3: Q78E (page 197)

Risk of a natural gas pipeline accident. Process Safety Progress (December 2004) published a risk analysis for a natural gas pipeline between Bolivia and Brazil. The most likely scenario for an accident would be natural gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (causing a jet fire) is .01. If the leak does not immediately ignite, it may result in a delayed ignition of a gas cloud. Given no immediate ignition, the probability of delayed ignition (causing a flash fire) is .01. If there is no ignition, the gas cloud will harmlessly disperse. Suppose a leak occurs in the natural gas pipeline. Find the probability that either a jet or flash fire will occur. Illustrate with a tree diagram.

Short Answer

Expert verified

The probability that either a jet fire or a flash fire will occur is 0.0199.

Step by step solution

01

Given information and definitions

P (jet fire) = 0.01

P (no jet fire) = 1-0.01 = 0.99

02

The final result and tree diagram

Use the multiplication rule, then

\(\begin{aligned}P\left( {{\rm{flash fire}}} \right) &= {\bf{ }}P\left( {{\rm{jet fire}}} \right) \times P\left( {{\rm{no jet fire}}} \right)\\ &= 0.01 \times 0.99\\ &= 0.0099\end{aligned}\)

\(\begin{aligned}P\left( {{\rm{jet fire or flash fire}}} \right) &= 0.01 + 0.0099\\ &= 0.0199\end{aligned}\)

Therefore, the result is\(0.0199\).

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Most popular questions from this chapter

Firefighter glove sizing. Human Factors (December 2015) published a study on how well firefighter gloves fit. In a group of 586 firefighters who reported their glove size, the researchers determined whether the gloves fit well or poorly by gender. The data are summarized in the accompanying table. Consider the gender and glove fit status of a randomly selected firefighter.

a. List the sample points for this experiment.

b. Assign reasonable probabilities to these sample points.

c. Find the probability the firefighter is a female.

d. Find the probability the glove fits well.

e. Find the probability the firefighter is a female and has a well-fitting glove.

f. Find the probability the firefighter is a female or has a well-fitting glove.

Glove Fits Well

Glove Fits Poorly

Totals

Males

415

132

547

Females

19

50

39

Totals

434

152

586

Source: H. Hsiao, et al., 鈥淔irefighter Hand Anthropometry and Structural Glove Sizing: A New Perspective,鈥� Human Factors, Vol. 57, No. 8, December 2015 (Table 6).

Consider the experiment depicted by the Venn diagram, with the sample space S containing five sample points. The sample points are assigned the following probabilities:

P (E1) = .20, P (E2) = .30, P (E3)= .30, P (E4) = .10, P (E5) = .10.

a. Calculate P (A), P (B), and P (A鈭�B).

b. Suppose we know that event A has occurred, so that the reduced sample space consists of the three sample points in A鈥攏amely, E1, E2, and E3. Use the formula for conditional probability to adjust the probabilities of these three sample points for the knowledge that A has occurred [i.e., P (Ei/A)]. Verify that the conditional probabilities are in the same proportion to one another as the original sample point probabilities.

c. Calculate the conditional probabilityP (E1/A)in two ways: (1) Add the adjusted (conditional) probabilities of the sample points in the intersection A鈭�B, as these represent the event that B occurs given that A has occurred; (2) use the formula for conditional probability:

P (B/A) =P (A鈭�B)P (A)

Verify that the two methods yield the same result.

d. Are events A and B independent? Why or why not?

Encoding variability in software. At the 2012 Gulf Petrochemicals and Chemicals Association (GPCA) Forum, Oregon State University software engineers presented a paper on modelling and implementing variation in computer software. The researchers employed the compositional choice calculus (CCC)鈥攁 formal language for representing, generating, and organizing variation in tree-structured artefacts. The CCC language was compared to two other coding languages鈥攖he annotative choice calculus (ACC) and the computational feature algebra (CFA). Their research revealed the following: Any type of expression (e.g., plain expressions, dimension declarations, or lambda abstractions) found in either ACC or CFA can be found in CCC; plain expressions exist in both ACC and CFA; dimension declarations exist in ACC, but not CFA; lambda abstractions exist in CFA, but not ACC. Based on this information, draw a Venn diagram illustrating the relationships among the three languages. (Hint: An expression represents a sample point in the Venn diagram.)

Problems at major companies. The Organization Development Journal (Summer 2006) reported on a survey of human resource officers (HROs) at major employers. The focus of the study was employee behaviour, namely, absenteeism and turnover. The study found that 55% of the HROs had problems with employee absenteeism; 41% had problems with turnover. Suppose that 22% of the HROs had problems with both absenteeism and turnover. Use this information to find the probability that an HRO selected from the group surveyed had problems with either employee absenteeism or employee turnover.

Are you really being served red snapper? Red snapper is arare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers

at the University of North Carolina used DNA analysis to examine fish specimens labeled 鈥渞ed snapper鈥� that were purchased from vendors across the country (Nature, July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper but the cheaper, look-alike variety of fish.

a. Assuming the results of the DNA analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant?

b. If there are five customers at a chain restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper?
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