Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Copykwik has four photocopy machines: \(A, B, C\), and \(D\). The probability that a given machine will break down on a particular day is \(P(A)=\frac{1}{50} \quad P(B)=\frac{1}{60} \quad P(C)=\frac{1}{75} \quad P(D)=\frac{1}{40}\) Assuming independence, what is the probability on a particular day that a. All four machines will break down? b. None of the machines will break down?

Step-by-Step Solution

Answer

a. The probability of all four machines breaking down on a particular day is \(\frac{1}{9,000,000}\). b. The probability of none of the machines breaking down on a particular day is \(\frac{135,341}{150,000}\).

Step 1: a. Probability of all four machines breaking down.

To find the probability of all machines breaking down, we will multiply the probabilities of each individual machine breaking down, as they are independent events. Let A, B, C, and D represent the events that machines A, B, C, and D break down, respectively. The probability that all four machines will break down is given by the product of their individual probabilities: \(P(A \cap B \cap C \cap D) = P(A)P(B)P(C)P(D)\) Plugging in the given probabilities: \(P(A \cap B \cap C \cap D) = (\frac{1}{50})(\frac{1}{60})(\frac{1}{75})(\frac{1}{40})\) Multiplying these probabilities gives: \(P(A \cap B \cap C \cap D) = \frac{1}{9,000,000}\) So, the probability of all four machines breaking down on a particular day is \(\frac{1}{9,000,000}\).

Step 2: b. Probability of none of the machines breaking down.

To find the probability of none of the machines breaking down, we will first find the probability of each individual machine not breaking down, and then multiply those probabilities together, as they are also independent events. The probability of an individual machine not breaking down is equal to 1 minus the probability of it breaking down. So, we have: \(P(A') = 1-P(A) = 1 - \frac{1}{50} = \frac{49}{50}\), \(P(B') = 1-P(B) = 1 - \frac{1}{60} = \frac{59}{60}\), \(P(C') = 1-P(C) = 1 - \frac{1}{75} = \frac{74}{75}\), \(P(D') = 1-P(D) = 1 - \frac{1}{40} = \frac{39}{40}\). Now, we multiply these probabilities together: \(P(A' \cap B' \cap C' \cap D') = P(A')P(B')P(C')P(D')\) \(= (\frac{49}{50})(\frac{59}{60})(\frac{74}{75})(\frac{39}{40})\) Multiplying these probabilities gives: \(P(A' \cap B' \cap C' \cap D') = \frac{135,341}{150,000}\) So, the probability of none of the machines breaking down on a particular day is \(\frac{135,341}{150,000}\).

Key Concepts

Independent Events

Imagine you're at Copykwik, a printing shop with four machines: A, B, C, and D. These machines work independently, meaning the functioning of one machine doesn't affect the others. In probability theory, when we talk about **independent events**, we mean that knowing one event has occurred doesn't tell us anything about the occurrence of another. For our machines, just because Machine A breaks down, it doesn't mean Machine B is more or less likely to break down. This is crucial for calculating probabilities. When events are independent, the probability of all events occurring together is simply the **product of their individual probabilities**. This simplifies calculations, as you don't need to consider interactions among events. For example, to find the probability of all four machines failing, you just multiply the probability of each one failing together. Remember, independence is a key assumption. If events were dependent, results would be quite different and more complex.

Complementary Probability

What if none of the machines breakdown at all? To figure that out, we use the idea of **complementary probabilities**. Complementary probability involves considering the opposite scenario of what we're interested in. For instance, if the probability of a machine breaking down is small, its non-breakdown probability (its complement) is large.For a single machine, say A, if the chance of breaking is \(\frac{1}{50}\), then the probability of **not** breaking is \(1 - \frac{1}{50} = \frac{49}{50}\). Apply this concept to each machine, and then multiply to find the probability of none breaking down since they are independent. This demonstrates how complementary probabilities provide a different perspective by focusing on what we don’t want to happen so that we can deduce what we do want.

Multiplication Rule for Probabilities

In probability, knowing how to combine events is essential. The **multiplication rule** helps us find the chance of a series of independent events all occurring. When you have several independent events and you want to know the likelihood they all happen, you multiply their probabilities. With our machines:- The chance that Machine A will fail is \(\frac{1}{50}\).- Machine B will fail with probability \(\frac{1}{60}\), and so on.Thus, to find the probability of all machines breaking down at the same time, you multiply these separate probabilities:\[P(A \cap B \cap C \cap D) = P(A) \cdot P(B) \cdot P(C) \cdot P(D)\]For four independent failures, multiply the probabilities together to get a tiny number, reflecting how unlikely it is. This rule simplifies calculating complex scenarios with several independent decisions or actions.