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Chapter 2: Problem 186

A machine tool is idle \(15 \%\) of the time. You request immediate use of the tool on five different occasions during the year. Assume that your requests represent independent events. (a) What is the probability that the tool is idle at the time of all of your requests? (b) What is the probability that the machine is idle at the time of exactly four of your requests? (c) What is the probability that the tool is idle at the time of at least three of your requests?

Short Answer

Expert verified
(a) 0.0000759, (b) 0.0032269, (c) 0.0389779.

Step by step solution

01

Understand the Problem

The machine tool is idle 15% of the time, meaning the probability of it being idle is \( p = 0.15 \). We have 5 requests which are independent events. We are asked to calculate the probability of different scenarios of the machine being idle during these requests.
02

Use Binomial Distribution

This problem can be solved using the binomial distribution, as we have a fixed number of independent trials (requests) with two possible outcomes (idle or not idle). The probability of exactly \( k \) successes (idle times) in \( n \) trials is given by the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \( n = 5 \) and \( p = 0.15 \).
03

Calculate Probability for All Idle Requests

For part (a), we need the probability that the machine is idle for all 5 requests, i.e., \( k = 5 \). Plug into the binomial formula:\[P(X = 5) = \binom{5}{5} (0.15)^5 (0.85)^0 = (0.15)^5 \]Calculate the result.
04

Calculate Probability for Exactly Four Idle Requests

For part (b), we calculate the probability that the machine is idle for exactly 4 requests, i.e., \( k = 4 \).\[P(X = 4) = \binom{5}{4} (0.15)^4 (0.85)^1 = 5 \times (0.15)^4 \times 0.85\]Calculate the result.
05

Calculate Probability for At Least Three Idle Requests

For part (c), 'at least three requests' means 3, 4, or 5 requests. We find the sum:\[P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5)\]Calculate \( P(X = 3) \) as:\[P(X = 3) = \binom{5}{3} (0.15)^3 (0.85)^2 = 10 \times (0.15)^3 \times (0.85)^2\]Sum all probabilities to find the answer.
06

Compute the Results

Using a calculator or software, compute the probabilities:- For \( P(X = 5) \), the probability is approximately 0.0000759.- For \( P(X = 4) \), the probability is approximately 0.0032269.- For \( P(X = 3) \), the probability is approximately 0.0356751.Add them up for part (c): \( P(X \geq 3) \approx 0.0389779 \).

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Key Concepts

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

Probability Theory
Probability theory is a fundamental concept in statistics that deals with the likelihood of events occurring. In our example, we look at the likelihood of a machine tool being idle. To determine such probabilities, we typically need to understand key elements such as:
  • Sample Space: All possible outcomes in an event, such as being idle or busy.
  • Event: A specific outcome we are interested in, for instance, the tool being idle.
  • Probability of an Event: It's a measure of the chance that the event will happen, usually represented between 0 and 1.
To solve our exercise, we leverage this theory to calculate how likely it is for a machine tool to be idle when requested five times. By determining the probability of the tool's idle state for each separate request, the overall probability of scenarios like all requests finding the tool idle can be assessed using a structured mathematical framework.
Independent Events
Independent events are those whose outcomes do not affect one another. In our machine tool scenario, each request's outcome is independent, meaning whether the tool is idle one time doesn鈥檛 affect future requests. Understanding independent events helps us utilize the binomial distribution because it ensures each trial or request is separate and unaffected by previous trials. In such cases:
  • The probability of an event remains constant. Here, the tool's idleness has a consistent 15% chance.
  • We can multiply the probability of outcomes across multiple events, which allows us to calculate situations like all requests being met with an idle tool.
  • The concept significantly simplifies complex probability calculations, turning them into a series of manageable steps.
Recognizing and applying the independent events concept is crucial when handling multiple trials like our five requests for the machine tool.
Success Probability Calculation
Success probability calculation in a binomial distribution represents how likely it is to achieve a specific number of successful events. In our situation, a success is when the machine tool is idle when requested. The binomial formula is key here:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
  • For 5 requests, where success probability (being idle) is 0.15, and failure is 0.85, this formula helps us calculate each specific outcome:
    • All 5 requests finding the machine idle: Plugging in values for \(k = 5\) delivers a tiny probability.
    • Exactly 4 idle requests: Involves calculating for \(k = 4\).
    • At least 3 idle requests: Adds results for \(k = 3, 4,\) and \(5\).
Using this framework allows for effective computation of complex and varied probability queries, as seen in our exercise.

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Most popular questions from this chapter

Six tissues are extracted from an ivy plant infested by spider mites. The plant in infested in \(20 \%\) of its area. Each tissue is chosen from a randomly selected area on the ivy plant. (a) What is the probability that four successive samples show the signs of infestation? (b) What is the probability that three out of four successive samples show the signs of infestation?

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