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Chapter 6: Problem 88

A certain company sends \(40 \%\) of its overnight mail parcels by means of express mail service \(A_{1}\). Of these parcels, \(2 \%\) arrive after the guaranteed delivery time (use \(L\) to denote the event late delivery). If a record of an overnight mailing is randomly selected from the company's files, what is the probability that the parcel went by means of \(A_{1}\) and was late?

Short Answer

Expert verified
The probability that a parcel sent by express mail service \(A_{1}\) arrives late is \(0.008\) or \(0.8\%\).

Step by step solution

01

Convert percentages to probabilities

First, convert the given percentages to equivalent probabilities. A percentage is simply a probability out of 100. Hence, 40% becomes \(0.40\) and 2% becomes \(0.02\).
02

Apply the multiplication rule of probability

To get the joint probability of two events, we multiply the probability of the first event by the probability of the second event given the first. This is called the multiplication rule. Here, the first event is that a parcel is sent by \(A_{1}\) and the second event is that it arrives late. So, we multiply these probabilities: \(0.40 * 0.02 = 0.008\).
03

Interpret the result

The result \(0.008\) is the probability that a parcel sent by \(A_{1}\) arrives late. In terms of percentage, this is \(0.8\%\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication Rule
Probability often involves calculating how likely it is for two events to happen together. This is where the multiplication rule comes in handy. Basically, it helps us figure out the joint probability of two events occurring at the same time.

For instance, if you want to find the probability that a parcel goes through a particular mail service and is late, you use the multiplication rule. It works by multiplying the probability of the first event (like choosing a specific mail service) by the probability of the second event given the first (such as the parcel arriving late if sent through that service).

Remember:
* Probability values are between 0 and 1.
* The closer a probability is to 0, the less likely the event is to happen.
* The closer it is to 1, the more likely the event is to happen.

So, the multiplication rule simplifies relationships between events by allowing us to combine their probabilities.
Joint Probability
Joint probability refers to the likelihood of two events occurring simultaneously. It's at the core of understanding how two or more independent or dependent events relate to one another in terms of probability.

In our example, we are curious about the joint probability that a parcel was sent via a certain mail service and arrived late. To calculate this, we use the multiplication rule, multiplying the probability of each event.

Here are a few important points about joint probability:
  • It combines separate probabilities into a link between events.
  • It requires the probabilities of all individual events involved.
  • It's essential for understanding dependencies between events.
Knowing how to compute joint probability is crucial for solving complex probability problems that involve several interconnected events.
Late Delivery Probability
Late delivery probability tells us the chance of a parcel not arriving on time. In scenarios where timely delivery is critical, understanding this probability is important for businesses and customers alike.

Sometimes, late deliveries might be due to factors beyond the control of a mail service, and customers need to know these odds to manage expectations.

For the company in our example, owning a record of their late delivery probability helps them improve their service and track performance. In this case, the late delivery probability for the parcels sent by their chosen mail service is only 2%, which seems low. However, multiplying this with another probability (using the multiplication rule) gives us the probability concerning specific scenarios, sometimes yielding unexpected insights.

Ultimately, assessing late delivery probabilities helps in planning logistical and customer satisfaction strategies.

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Most popular questions from this chapter

Five hundred first-year students at a state university were classified according to both high school GPA and whether they were on academic probation at the end of their first semester. The data are $$ \begin{array}{lcccc} && {\text { High School GPA }} \\ & 2.5 \text { to } & 3.0 \text { to } & 3.5 \text { and } & \\ \text { Probation } & <3.0 & <3.5 & \text { Above } & \text { Total } \\ \hline \text { Yes } & 50 & 55 & 30 & 135 \\ \text { No } & 45 & 135 & 185 & 365 \\ \text { Total } & 95 & 190 & 215 & 500 \\ \hline \end{array} $$ a. Construct a table of the estimated probabilities for each GPA-probation combination. b. Use the table constructed in Part (a) to approximate the probability that a randomly selected first-year student at this university will be on academic probation at the end of the first semester. c. What is the estimated probability that a randomly selected first-year student at this university had a high school GPA of \(3.5\) or above? d. Are the two outcomes selected student has a bigh school GPA of \(3.5\) or above and selected student is on academic probation at the end of the first semester independent outcomes? How can you tell? e. Estimate the proportion of first-year students with high school GPAs between \(2.5\) and \(3.0\) who are on academic probation at the end of the first semester. f. Estimate the proportion of those first-year students with high school GPAs \(3.5\) and above who are on academic probation at the end of the first semester.

A student placement center has requests from five students for interviews regarding employment with a particular consulting firm. Three of these students are math majors, and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students. These two will be randomly selected from among the five. a. What is the probability that both selected students are statistics majors? b. What is the probability that both students are math majors? c. What is the probability that at least one of the students selected is a statistics major? d. What is the probability that the selected students have different majors?

The article "Birth Beats Long Odds for Leap Year Mom, Baby" (San Luis Obispo Tribune, March 2 , 1996) reported that a leap year baby (someone born on February 29) became a leap year mom when she gave birth to a baby on February 29,1996 . The article stated that a hospital spokesperson said that only about 1 in \(2.1\) million births is a leap year baby born to a leap year mom (a probability of approximately .00000047). a. In computing the given probability, the hospital spokesperson used the fact that a leap day occurs only once in 1461 days. Write a few sentences explaining how the hospital spokesperson computed the stated probability. b. To compute the stated probability, the hospital spokesperson had to assume that the birth was equally likely to occur on any of the 1461 days in a four- year period. Do you think that this is a reasonable assumption? Explain. c. Based on your answer to Part (b), do you think that the probability given by the hospital spokesperson is too small, about right, or too large? Explain.

A company uses three different assembly lines \(A_{1}, A_{2}\), and \(A_{3}\) -to manufacture a particular component. Of those manufactured by \(A_{1}, 5 \%\) need rework to remedy a defect, whereas \(8 \%\) of \(A_{2}\) 's components and \(10 \%\) of \(A_{3}\) 's components need rework. Suppose that \(50 \%\) of all components are produced by \(A_{1}\), whereas \(30 \%\) are produced by \(A_{2}\) and \(20 \%\) come from \(A_{3}\). a. Construct a tree diagram with first-generation branches corresponding to the three lines. Leading from each branch, draw one branch for rework (R) and another for no rework (N). Then enter appropriate probabilities on the branches. b. What is the probability that a randomly selected component came from \(A_{1}\) and needed rework? c. What is the probability that a randomly selected component needed rework?

There are five faculty members in a certain academic department. These individuals have \(3,6,7,10\), and 14 years of teaching experience. Two of these individuals are randomly selected to serve on a personnel review committee. What is the probability that the chosen representatives have a total of at least 15 years of teaching experience? (Hint: Consider all possible committees.)
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