Problem 216 - Applied Statistics and Probability for Engineers

Question

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?

Step-by-Step Solution

Answer

a) 0.000076. b) 0.002153. c) 0.026667.

Step 1: Identify Probability of a Single Idle Event

The tool is idle 15% of the time. Therefore, the probability that the tool is idle when you request it once is given by \( P(I) = 0.15 \), where \( I \) denotes a single idle event.

Step 2: Determine Probability of All Idle Requests

To find the probability that the tool is idle at the time of all five requests, we use the formula for independent events: \( P(I^5) = P(I) \times P(I) \times P(I) \times P(I) \times P(I) = (0.15)^5 \). Calculating: \( (0.15)^5 = 7.59375 \times 10^{-5} \).

Step 3: Find Probability for Exactly Four Idle Requests

Use the binomial probability formula: \( P(X=k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \), where \( n=5 \), \( k=4 \), and \( p=0.15 \). So, \( P(X=4) = \binom{5}{4} \times (0.15)^4 \times (1-0.15)^{5-4} \). Calculate \( \binom{5}{4} = 5 \) and \( P(X=4) = 5 \times (0.15)^4 \times (0.85) = 5 \times 0.00050625 \times 0.85 = 0.002153 \).

Step 4: Calculate Probability for At Least Three Idle Requests

To find the probability that the tool is idle at least three times, sum the probabilities for 3, 4, and 5 idle responses. \( P(X \geq 3) = P(X=3) + P(X=4) + P(X=5) \). \( P(X=3) = \binom{5}{3} \times (0.15)^3 \times (0.85)^{5-3} \). Calculate \( \binom{5}{3} = 10 \) and \( P(X=3) = 10 \times (0.15)^3 \times (0.85)^2 = 10 \times 0.003375 \times 0.7225 = 0.024439 \). Aggregate: \( P(X \geq 3) = 0.024439 + 0.002153 + 7.59375 \times 10^{-5} = 0.026667 \).

Key Concepts

Binomial Distribution

Binomial distribution is a key concept in probability theory that deals with experiments having two possible outcomes: success or failure. It is particularly useful when the number of trials is fixed, each trial is independent of the others, and the probability of success is the same in each trial.
For example, when determining the probability of a machine being idle on exactly five requests, a binomial distribution can be employed.

• Each request can be seen as a "trial".
• The machine being idle is a "success" with a probability of 0.15.
• The binomial distribution formula, given by \( P(X=k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \), helps calculate specific outcomes.

In the exercise, we calculate the probability of exactly four idle times, considering five requests, to understand how this distribution plays out in practical scenarios. This helps elucidate the nature of probability spread across discrete events.

Independent Events

Independent events are a core concept of probability theory, vital for calculating compound probabilities. When two or more events do not affect each other, they are said to be independent. This means the occurrence of one event has no impact on the occurrence of another.

• In the given problem, each request to use the tool is an independent event.
• The probability that the machine is idle remains constant, regardless of past or future requests.
• This independence ensures that probabilities can be multiplied to find the likelihood of multiple events occurring in sequence.

Thus, when calculating the probability of the tool being idle on all five requests, you use this property, multiplying the individual probabilities: \((0.15)^5\). This fundamental principle simplifies calculations and highlights how separate events do not influence each other unless stated otherwise.

Idle Time Probability

Idle time probability refers to the chances of a system being unoccupied or not in use at a given moment. This concept is critical in fields like operations management and production planning, where maximizing machine usage and minimizing downtime is often a priority.
In the problem scenario:

• The machine's idle probability is given as 15% (or 0.15).
• Understanding this probabilistic measure allows for foreseeing system inefficiencies or improving scheduling.
• This concept can be extended to different time frames or conditions, like cumulative probabilities over multiple occurrences.

The calculations in the exercise – such as determining the tool’s idle likelihood at various times – emphasize the practical use of idle time probability to anticipate and manage real-world outcomes effectively. Knowing this probability also aids decision-making in both logistics and strategic planning across operations.