Problem 11 - Applied Probability

Question

Two software companies, MicroWare and BusiCorp, are preparing a new business package in time for a computer trade show 180 days in the future. They work independently. MicroWare has anticipated completion time, in days, exponential (1/150). BusiCorp has time to completion, in days, exponential \((1 / 130)\). What is the probability both will complete on time; that at least one will complete on time; that neither will complete on time?

Step-by-Step Solution

Answer

The probabilities are as follows: both complete on time: \(1 - e^{-1.2})\times(1 - e^{-1.3846})\), at least one completes on time: \(1 - e^{-1.2}\times e^{-1.3846})\), neither completes on time: \(e^{-1.2} \times e^{-1.3846})\).

Step 1: Understanding the Exponential Distribution

MicroWare and BusiCorp have completion times that are exponentially distributed with means of 150 and 130 days, respectively. The probability density function (PDF) of an exponential distribution with the mean \(\mu\) is \(f(t) = \frac{1}{\mu} e^{-t/\mu}\) where \(t\) is time. For MicroWare, with \(\mu = 150\), the PDF is \(f(t) = \frac{1}{150} e^{-t/150}\), and for BusiCorp, with \(\mu = 130\), the PDF is \(f(t) = \frac{1}{130} e^{-t/130}\).

Step 2: Probability of Completion On Time for MicroWare

To find the probability that MicroWare will finish on time (within 180 days), calculate the cumulative distribution function (CDF) which is \(1 - e^{-t/\mu}\). For MicroWare, it's \(1 - e^{-180/150} = 1 - e^{-1.2}\).

Step 3: Probability of Completion On Time for BusiCorp

Similarly for BusiCorp, calculate the CDF for an on time finish within 180 days using their rate, which is \(1 - e^{-180/130} = 1 - e^{-1.3846}\).

Step 4: Probability Both Companies Complete On Time

Since the companies work independently, the joint probability of both completing on time is the product of their individual probabilities. Multiply the results from the CDFs of MicroWare and BusiCorp to find the joint probability.

Step 5: Probability At Least One Completes On Time

This probability is equal to 1 minus the probability that neither company completes on time. To find the probability that neither completes on time, multiply the probabilities that each company will not complete on time (the complement of their individual CDFs).

Step 6: Probability Neither Completes On Time

To find this probability, calculate the product of the complements of their individual CDFs: \(e^{-180/150} \cdot e^{-180/130}\).

Step 7: Computing the Probabilities

Insert the calculated values into the formulas from the previous steps to find the actual numeric probabilities for each scenario.

Key Concepts

Probability Density Function (PDF)

The Probability Density Function (PDF) is a fundamental concept in statistics, particularly for continuous random variables such as the time it takes for a software company to complete a project. In the context of exponential distribution, the PDF is defined as \(f(t) = \frac{1}{\mu} e^{-t/\mu}\), where \(t\) represents time and \(\mu\) is the mean completion time.

The PDF helps us understand how the probability distribution of time to completion is spread out over time. For a company with a mean completion time of \(\mu\), the PDF will peak at 0 and decay exponentially as time increases, indicating that shorter completion times are more likely than longer ones. This exponential decay is captured by the term \(e^{-t/\mu}\) in the formula.

Understanding the PDF is crucial because it serves as the basis for calculating the likelihood of completing a project within a specific timeframe, which in our case, is represented by the exercise question involving MicroWare and BusiCorp.

Cumulative Distribution Function (CDF)

While the PDF provides the probability of a precise value, the Cumulative Distribution Function (CDF) provides the probability that a random variable is less than or equal to a certain value. For the exponential distribution, the CDF is expressed as \(1 - e^{-t/\mu}\).

The CDF is used to determine the probability of a software company completing its project on or before a certain day. It is essentially the accumulation of probabilities from 0 to the specified time \(t\). In practical terms, computing the CDF for MicroWare and BusiCorp gives us their respective probabilities of completing the project within 180 days.

Since these probabilities are cumulative, they increase as time passes, up to a point where they eventually reach 1. The application of CDF in our exercise allows us to predict the likelihood that the companies will meet their deadlines, an essential factor in project planning and management.

Independent Events Probability

When we deal with independent events, the probability of multiple events occurring together is found by multiplying their individual probabilities. Independence means the outcome of one event does not affect the outcome of another.

In our example, MicroWare and BusiCorp work independently, which means the completion time of one does not influence the other. Therefore, to calculate the likelihood of both companies finishing on time, we multiply the CDF results for each. Conversely, the probability of at least one company finishing on time involves a different approach—we consider the probability of neither company finishing on time and subtract it from 1.

This concept is vital in project management and risk assessment where multiple independent tasks or project phases can dictate the overall success. By understanding how to calculate probabilities of independent events, one can adequately estimate completion risks and set realistic timelines for project deliverables.