Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The proprietor of Cunningham's Hardware Store has decided to install floodlights on the premises as a measure against vandalism and theft. If the probability is \(.01\) that a certain brand of floodlight will burn out within a year, find the minimum number of floodlights that must be installed to ensure that the probability that at least one of them will remain functional for the whole year is at least .99999. (Assume that the floodlights operate independently.)

Step-by-Step Solution

Answer

The minimum number of floodlights needed to ensure that the probability of at least one of them remaining functional for the whole year is at least .99999 is \(n = 2301\).

Step 1: Define the given probability

We are given the probability, \(p\), that a certain brand of floodlight will burn out within a year. In this case, \(p = 0.01\).

Step 2: Use the complementary probability

Since we want to find the probability that at least one floodlight remains functional for the entire year, we will first find the probability that none of the floodlights remain functional for the entire year. The probability that a single floodlight remains functional for the entire year is \(1-p = 1-0.01 = 0.99\).

Step 3: Calculate the probability based on the number of floodlights

Let's say we have \(n\) floodlights installed. Since the floodlights operate independently, the probability that all \(n\) floodlights will burn out within a year is \((0.99)^n\).

Step 4: Subtract the calculated probability from 1 to find the probability of at least one remaining functional

The probability that at least one floodlight remains functional for the entire year is the opposite of the probability that all \(n\) floodlights will burn out, so we find this value by subtracting the probability found in step 3 from 1: \(1 - (0.99)^n\).

Step 5: Set the function equal to the desired probability and solve for the minimum number of floodlights

We want the probability that at least one floodlight remains functional for the entire year to be at least .99999. Therefore, we set up the equation: \(1 - (0.99)^n \geq 0.99999\). In order to find the minimum number of floodlights needed, we have to solve this inequality for \(n\). To solve this inequality, first subtract 1 from both sides: \(-(0.99)^n \geq -0.00001\). Next, divide both sides of the inequality by -1, which will change the direction of the inequality: \((0.99)^n \leq 0.00001\). Now, take the natural logarithm of both sides to solve for \(n\): \(n \cdot \ln(0.99) \leq \ln(0.00001)\) Finally, divide both sides by \(\ln(0.99)\) to isolate \(n\): \(n \geq \frac{\ln(0.00001)}{\ln(0.99)}\approx 2300.30\). Since \(n\) must be a whole number, the minimum number of floodlights needed to ensure that the probability of at least one of them remaining functional for the whole year is at least .99999 is \(n=\lceil2300.30\rceil = 2301\).

Key Concepts

Independent Events

Independent events are a fundamental concept in probability, helping us understand how different occurrences relate to each other. When two events are independent, this means that the outcome of one event does not influence the outcome of another.

• In the case of the floodlights, each light burning out within a year is independent of another light's performance.
• This independence allows us to multiply probabilities of individual events to find a joint probability.

Understanding the independence of events is crucial when evaluating scenarios where multiple actions might occur simultaneously. In Cunningham's Hardware Store challenge, we relied on the principle of independent events to compute the probability that all floodlights would fail in order to find out how many must stay functional.

Complementary Probability

Complementary probability is a concept that helps us determine the chance of an event not occurring, by utilizing the probability of the event occurring. The idea revolves around the notion that the sum of probabilities for an event occurring and not occurring is always 1.

• For example, if the probability of a floodlight burning out within a year is 0.01, the complementary probability — it lasting a year — is 1 - 0.01 = 0.99.
• This relationship is key when calculating the total probability, especially in finding the probability of at least one success in multiple trials.

In the exercise, we used the complementary probability to understand how likely it is that not all floodlights would burn out over the course of a year.

Inequalities in Mathematics

Inequalities are mathematical expressions that represent the relative size or order of two values without showing equality.

• In this context, inequalities were used to set up an equation to solve for the number of floodlights that must work to reach a certain probability.
• The inequality \(1 - (0.99)^n \geq 0.99999\) highlights the relationship between the probability of at least one working floodlight and the total number installed.

By incorporating inequalities, we can determine necessary conditions to meet specific probability outcomes, making inequalities an essential tool when solving problems with desired constraints.

Natural Logarithm

The natural logarithm is a mathematical function denoted as \(\ln\), which is the inverse of the exponential function involving Euler's number \(e\approx 2.71828\).

• It is used to solve equations where variables are exponents, as seen when determining the number of floodlights necessary in this problem.
• Taking the logarithm of both sides of the equation \((0.99)^n \leq 0.00001\) allowed us to move the exponent down, making it simpler to solve for \(n\).

In this solution, the natural logarithm facilitated the conversion of multiplicative processes into additive ones, making it easier to manipulate and solve the problem step by step. Understanding the natural logarithm's role is vital for tackling complex equations and inequalities in probability and statistics.