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Chapter 2: Problem 25

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) 0.000077 b) 0.003018 c) 0.03549125

Step by step solution

01

Understand the basic probability

The probability that the machine tool is idle at any given time is 15%, or 0.15, and the probability that the machine is not idle is 85%, or 0.85.
02

Use the binomial probability formula

Since each request is an independent Bernoulli trial, we use the binomial distribution to find probabilities. The formula for binomial probability is \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the number of trials, \( k \) is the number of successful trials, \( p \) is the probability of success, and \( \binom{n}{k} \) is the binomial coefficient.
03

Calculate the probability for all requests idle (a)

Let \( n = 5 \), \( k = 5 \), and \( p = 0.15 \). Apply these values to the binomial formula: \[ P(X = 5) = \binom{5}{5} (0.15)^5 (0.85)^0 = (0.15)^5 \approx 0.0000765 \].Thus, the probability is approximately 0.000077.
04

Calculate the probability for exactly four requests idle (b)

Let \( n = 5 \), \( k = 4 \), and \( p = 0.15 \). Apply these values to the binomial formula: \[ P(X = 4) = \binom{5}{4} (0.15)^4 (0.85)^1 = 5 \times (0.15)^4 \times (0.85) \approx 0.003018 \].Thus, the probability is approximately 0.003018.
05

Calculate the probability for at least 3 requests idle (c)

We find the probability of 3 idle requests and add it to the probabilities of 4 and 5 idle requests. Calculate for \( k = 3 \):\[ P(X = 3) = \binom{5}{3} (0.15)^3 (0.85)^2 = 10 \times (0.15)^3 \times (0.85)^2 \approx 0.03239625 \].Then, the total probability: \[ P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5) \approx 0.03239625 + 0.003018 + 0.000077 \approx 0.03549125 \].

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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 the branch of mathematics that deals with the likelihood of different events occurring. In our example, it helps us figure out if the machine tool will be idle when we need it. Let's break it down simply:
  • The probability of any event happening is a number between 0 and 1.
  • If the probability is 0, the event will not happen. If it's 1, it will definitely happen.
  • The sum of probabilities of all possible outcomes of a random experiment is always 1.
For instance, if there's a 15% chance the machine is idle, it translates to a probability of 0.15. Likewise, if it's not idle, the probability would be 0.85 (since 1 - 0.15 = 0.85). Pretty straightforward, right?
Probability makes it easier to predict how likely an event is, which is essential when you have multiple such independent events throughout the year.
Independent Events
In probability, independent events are those where the outcome of one does not affect the outcome of the other. In our context, each time you request the tool is considered an independent event. Here's why:
  • The machine's idleness during one request doesn't change its idleness status during another request.
  • Even if your tool is available the first time, it doesn't mean it will be available or not available the next time.
This concept is crucial because it means we can treat each request separately when calculating probabilities. It simplifies our calculations, as we don鈥檛 have to consider past outcomes when evaluating future ones.
Understanding independence is key to applying the binomial distribution correctly in our scenario. It ensures that each request is seen as separate, making our probability calculations both accurate and efficient.
Bernoulli Trials
A Bernoulli trial is an experiment or process that results in a binary outcome: success or failure. Each request in our problem can be seen as a Bernoulli trial. Here's what you need to know:
  • "Success" in this context means finding the tool idle, with a probability of 0.15.
  • "Failure" would be the tool being busy, with a probability of 0.85.
  • Every request is an independent Bernoulli trial, with the same probability of success (0.15).
Bernoulli trials are foundational to understanding binomial distribution, as a binomial setup involves multiple Bernoulli trials. The simplicity of trials being only success or failure helps streamline complex calculations in probability.
This kind of trial offers a clear framework to examine each chance of the machine being idle independently, thus aligning perfectly with our usage of the binomial distribution to find the probability of multiple requests.
Binomial Coefficient
The binomial coefficient is a key part of the binomial formula used to calculate the probability of success in a series of Bernoulli trials. It's represented as \( \binom{n}{k} \), pronounced as "n choose k." Let's dive into what it means:
  • \( n \) is the total number of trials, which is 5 in our example.
  • \( k \) is the number of successful trials you're interested in.
  • The binomial coefficient \( \binom{n}{k} \) tells you how many ways you can achieve exactly \( k \) successes in \( n \) trials.
For instance, \( \binom{5}{3} \) represents the number of ways to have 3 successes out of 5 trials. It's calculated using the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]where "!" denotes factorial, the product of all positive integers up to that number.
Understanding the binomial coefficient is crucial because it lets us calculate probabilities accurately when applying the binomial distribution to independent Bernoulli trials.

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Most popular questions from this chapter

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