Problem 36 - Algebra 2

Question

UTILITIES A city water system includes a sequence of 4 pumps as shown below. Water enters the system at point \(A,\) is pumped through the system by pumps at locations \(1,2,3,\) and \(4,\) and exits the system at point \(B .\) If the probability of failure for any one pump is \(\frac{1}{100},\) what is the probability that water will flow all the way through the system from A to \(\mathrm{B} ?\)

Step-by-Step Solution

Answer

The probability that water will flow from A to B is \(\left(\frac{99}{100}\right)^4\).

Step 1: Understand the Problem

We have 4 pumps in sequence, and we need to find the probability that all pumps work without failure for water to flow from A to B. Each pump has a failure probability of \(\frac{1}{100}\).

Step 2: Determine the Probability of Success for Each Pump

If the probability of failure for each pump is \(\frac{1}{100}\), then the probability of a pump not failing (success) is \(1 - \frac{1}{100} = \frac{99}{100}\).

Step 3: Calculate the Probability of Success for All Pumps

The water will flow through successfully only if all four pumps work. The probability of all pumps working sequentially is the product of their individual success probabilities: \(\left(\frac{99}{100}\right)^4\).

Step 4: Perform the Calculation

Compute the power: \(\left(\frac{99}{100}\right)^4 = \frac{99}{100} \times \frac{99}{100} \times \frac{99}{100} \times \frac{99}{100}\). Calculate it to find the probability that water flows all the way from A to B.

Key Concepts

Failure Rate

In probability and statistics, understanding the concept of failure rate is crucial, especially when evaluating systems that involve various components working in sequence. The failure rate is defined as the probability that a component or system will fail during a specified time period. In the context of our exercise, each pump in the water system has a failure probability of \( \frac{1}{100} \). This means that there is a 1% chance that any individual pump will fail.

The failure rate is a fundamental concept because it helps in determining the reliability of a complex system. With a known failure rate, we can calculate how likely a system is to function without failure over a given period. Recognizing the failure rate enables us to navigate and mitigate risks associated with system performance.

Sequential Probability

Sequential probability comes into play when evaluating events that occur in a sequence, such as the water passing through multiple pumps in our exercise. The success or failure of each component in the sequence affects the outcome of the entire process.

When dealing with sequential probability, each event's probability is considered in order, as they occur one after another. For our water system, the probability of water flowing successfully from point A to point B depends on each of the four pumps working consecutively without failure.

Calculating the sequential probability involves multiplying the success probabilities of each pump, as each step in the sequence contributes to the final probability outcome.

Probability of Independent Events

The concept of independent events is pivotal in probability calculations involving multiple components, such as the pumps in our system. Independent events are those whose outcomes are not affected by one another. In this context, the performance of one pump does not influence the performance of the others.

When calculating the probability of all independent events occurring successfully, we multiply their individual probabilities. For our water system, we assume each pump operates independently, with a probability of success at \( \frac{99}{100} \). Therefore, the overall probability is calculated by multiplying the success probabilities of all four pumps: \((\frac{99}{100})^4\).

Understanding the independence of events helps simplify complex probability calculations by enabling us to treat each event as a separate entity.

Probability Calculation Steps

Calculating probabilities, especially in sequences like the water system in our exercise, involves clear steps. Here’s how we can think about this calculation process:

• Identify the probability of each event's success or failure. Here, each pump has a failure probability of \( \frac{1}{100} \) and a success rate of \( \frac{99}{100} \).
• Determine if the events are independent. In our system, each pump operates without influencing the others, making them independent events.
• Multiply the success probabilities of each independent event to find the total probability of success. This is done using the formula \((\frac{99}{100})^4\) for four pumps.
• Perform the calculation to find the final probability, indicating whether or not the water flows all the way through the system.

Following these steps ensures a structured approach to solving probability problems, especially those involving sequences of independent events. This clarity in calculation is key for accurate, reliable results.