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Chapter 5: Problem 93

Spoke Weaving Corporation has eight weaving machines of the same kind and of the same age. The probability is .04 that any weaving machine will break down at any time. Find the probability that at any given time a. all eight weaving machines will be broken down b. exactly two weaving machines will be broken down c. none of the weaving machines will be broken down

Short Answer

Expert verified
The probability that all eight machines break down is calculated in Step 2, that exactly two machines break down is calculated in Step 3, and that no machine breaks down is calculated in Step 4.

Step by step solution

01

Understanding the Binomial Probability Formula

The formula to calculate the binomial probability is \( P(X=k) = C(n, k) \cdot (p)^k \cdot (1-p)^{n-k} \), where P(X=k) is the probability of k successes in n trials, C(n, k) is the number of combinations of n items taken k at a time, p is the probability of success on any given trial, and (1-p) is the probability of failure.
02

Calculating the Probability of All Machines Breaking Down

We need to calculate the probability that all eight machines break down (k=8). Substitute n=8, k=8, and p=0.04 into the formula: \( P(8) = C(8, 8) \cdot (0.04)^8 \cdot (1-0.04)^{8-8} \). Simplifying this gives the required probability.
03

Calculating the Probability of Exactly Two Machines Breaking Down

This scenario involves exactly two machines breaking down (k=2). Substitute n=8, k=2, and p=0.04 into the formula: \( P(2) = C(8, 2) \cdot (0.04)^2 \cdot (1-0.04)^{8-2} \). Simplifying this gives the required probability.
04

Calculating the Probability of No Machines Breaking Down

This scenario involves no machines breaking down (k=0). Substitute n=8, k=0, and p=0.04 into the formula: \( P(0) = C(8, 0) \cdot (0.04)^0 \cdot (1-0.04)^{8-0} \). Simplifying this gives the required probability.

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Key Concepts

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

Combinations
When dealing with binomial probability, understanding combinations is essential. Combinations help us determine the number of ways we can choose a certain number of items from a larger set, without considering the order in which they're chosen.

In mathematical terms, the number of combinations of n items taken k at a time is denoted as \( C(n, k) \), which can be calculated using the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] Here, \( n! \) ("n factorial") represents the product of all positive integers up to n, making it \( n \times (n-1) \times (n-2) \times \ldots \times 1 \). Calculating combinations correctly is crucial for applying the binomial probability formula.
  • For example, \( C(8, 2) \) calculates how many ways we can choose 2 machines to break down from 8.
Probability of Success
The probability of success refers to the likelihood of a desired outcome in a single trial. In the context of our exercise, a 'success' is when a weaving machine breaks down.

The probability of success (\( p \)) can greatly influence the overall outcome in a binomial distribution. In this problem, each machine has a probability of 0.04 to break down, which we consider as our probability of success.
  • Using the formula: \( P(X=k) = C(n, k) \times (p)^k \times (1-p)^{n-k} \), the \( p^k \) term represents the probability of having \( k \) successes out of \( n \) trials.
  • For instance, when computing the probability that exactly two machines will break down, \( p^2 = (0.04)^2 \).
Probability of Failure
The probability of failure is simply the complementary probability of success for each trial. It denotes the likelihood of not achieving the desired outcome. In our case, it's the chance that a weaving machine does not break down during a given period.

To find this, we subtract the probability of success from 1: \( 1 - p \). This gives us a probability of \( 0.96 \) for a machine not breaking down.
  • The formula: \( P(X=k) = C(n, k) \times (p)^k \times (1-p)^{n-k} \) uses \( (1-p)^{n-k} \) to calculate the probability of \( n-k \) failures.
  • For example, if we want the probability that no machines break down, we calculate \( (0.96)^8 \).
Binomial Distribution
A binomial distribution is used to model the probability of obtaining a fixed number of successes in a set number of independent trials, each with the same probability of success.

It is characterized by two parameters: * The number of trials (denoted as \( n \)), which in our problem is 8, the number of machines.
* The probability of success (denoted as \( p \)), which is 0.04 in our scenario.

The binomial distribution is an essential concept in probability and statistics because it allows for the calculation of the likelihood of various outcomes. We use the formula: \( P(X=k) = C(n, k) \times (p)^k \times (1-p)^{n-k} \), to compute probabilities for different numbers of machine breakdowns.
  • For example, using this distribution, we can find the probability of exactly zero, two, or all eight machines breaking down.
Understanding how to correctly apply the binomial distribution is key in solving exercises like this.

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Most popular questions from this chapter

Residents in an inner-city area are concerned about drug dealers entering their neighborhood. Over the past 14 nights, they have taken turns watching the street from a darkened apartment. Drug deals seem to take place randomly at various times and locations on the street and average about three per night. The residents of this street contacted the local police, who informed them that they do not have sufficient resources to set up surveillance. The police suggested videotaping the activity on the street, and if the residents are able to capture five or more drug deals on tape, the police will take action. Unfortunately, none of the residents on this street owns a video camera and, hence, they would have to rent the equipment. Inquiries at the local dealers indicated that the best available rate for renting a video camera is \(\$ 75\) for the first night and \(\$ 40\) for each additional night. To obtain this rate, the residents must sign up in advance for 芒 specified number of nights. The residents hold a neighborhood meeting and invite you to help them decide on the length of the rental period. Because it is difficult for them to pay the rental fees, they want to know the probability of taping at least five drug deals on a given number of nights of videotaping. a. Which of the probability distributions you have studied might be helpful here? b. What assumption(s) would you have to make? c. If the residents tape for two nights, what is the probability they will film at least five drug deals? d. For how many nights must the camera be rented so that there is at least \(.90\) probability that five or more drug deals will be taped?

A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are \(\$ 10\) million from the office building, \(\$ 5\) million from the theater, and \(\$ 2\) million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are \(15, .30, .45\), and \(.10\), respectively. Let \(x\) be the random variable that represents the contractor's profits in millions of dollars. Write the probability distribution of \(x\). Find the mean and standard deviation of \(x\). Give a brief interpretation of the values of the mean and standard deviation.

An instant lottery ticket costs \(\$ 2 .\) Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of \(\$ 5\) each, 100 tickets have a prize of \(\$ 10\) each, 5 tickets have a prize of \(\$ 1000\) each, and 1 ticket has a prize of \(\$ 5000\). Let \(x\) be the random variable that denotes the net amount a player wins by playing this lottery. Write the probability distribution of \(x\). Determine the mean and standard deviation of \(x\). How will you interpret the values of the mean and standard deviation of \(x\) ?

The number of calls that come into a small mail-order company follows a Poisson distribution. Currently, these calls are serviced by a single operator. The manager knows from past experience that an additional operator will be needed if the rate of calls exceeds 20 per hour. The manager observes that 9 calls came into the mail-order company during a randomly selected 15 -minute period. a. If the rate of calls is actually 20 per hour, what is the probability that 9 or more calls will come in during a given 15 -minute period? b. If the rate of calls is really 30 per hour, what is the probability that 9 or more calls will come in during a given 15 -minute period? c. Based on the calculations in parts a and \(\mathrm{b}\), do you think that the rate of incoming calls is more likely to be 20 or 30 per hour? d. Would you advise the manager to hire a second operator? Explain.

An office supply company conducted a survey before marketing a new paper shredder designed for home use. In the survey, \(80 \%\) of the people who used the shredder were satisfied with it. Because of this high acceptance rate, the company decided to market the new shredder. Assume that \(80 \%\) of all people who will use it will be satisfied. On a certain day, seven customers bought this shredder. a. Let \(x\) denote the number of customers in this sample of seven who will be satisfied with this shredder. Using the binomial probabilities table (Table I, Appendix C), obtain the probability distribution of \(x\) and draw a graph of the probability distribution. Find the mean and standard deviation of \(x\). b. Using the probability distribution of part a, find the probability that exactly four of the seven customers will be satisfied.
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