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Chapter 2: Problem 15

An integrated circuit contains 10 million logic gates (each can be a logical AND or OR circuit). Assume the probability of a gate failure is \(p\) and that the failures are independent. The integrated circuit fails to function if any gate fails. Determine the value for \(p\) so that the probability that the integrated circuit functions is \(0.95 .\)

Short Answer

Expert verified
The probability of gate failure \( p \) is approximately \( 5.13 \times 10^{-9} \).

Step by step solution

01

Understand the Problem

The integrated circuit contains 10 million logic gates, and the circuit fails if any of these gates fails. We need to find the probability of a gate failure, denoted as \( p \), so that the probability that the entire circuit functions is 0.95.
02

Express Circuit Function Probability

Since the circuit functions only if all gates work properly, the probability of the circuit functioning can be expressed as \( (1-p)^{10^7} = 0.95 \). This is because each gate has a probability \( 1-p \) of functioning, and there are \( 10^7 \) such gates in the circuit.
03

Solve for Probability of Gate Failure (\( p \))

To find \( p \), solve the equation \( (1-p)^{10^7} = 0.95 \). First take the natural logarithm of both sides to get \( 10^7 \ln(1-p) = \ln(0.95) \). Then solve for \( \ln(1-p) \) by dividing both sides by \( 10^7 \).
04

Compute \( 1-p \) and \( p \)

Calculate \( \ln(1-p) = \frac{\ln(0.95)}{10^7} \). Exponentiate both sides to find \( 1-p \), giving \( 1-p = e^{\frac{\ln(0.95)}{10^7}} \). Finally, solve for \( p \) by subtracting the result from 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability, events are termed 'independent' when the outcome of one event does not affect the other. This concept is crucial in many engineering applications, especially when assessing reliability and risk of complex systems.
For example, consider a series of logic gates within an integrated circuit. Every gate functions independently鈥攎eaning one gate's success or failure doesn't influence any others.
  • When assessing reliability, it's important to know that the probability of all independent events occurring is the product of their individual probabilities.
  • If the gates in our circuit fail independently, the overall system's reliability is determined multiplicatively.
This is foundational for calculating the probability of failure or success in the entire system.
Probability of Failure
The probability of failure is a key metric in engineering, particularly when dealing with critical systems. It quantifies the likelihood that a component or system will fail to perform its intended function.
In our integrated circuit example, if the probability of a single gate failing is \(p\), the task is to make sure that the whole circuit鈥攚hich fails if any single gate fails鈥攆unctions as intended.
  • If even one gate fails, the entire system fails. Thus, the probability that the circuit functions is \((1-p)^{n}\), where \(n\) is the number of gates.
  • In practice, reducing \(p\) can significantly improve system reliability.
By understanding these probabilities, engineers can design more resilient systems.
Exponential Function
An exponential function is a mathematical expression in the form of \(a^x\). In probability and reliability calculations, these functions describe how a quantity grows or decays exponentially.
In our exercise, the function \((1-p)^{10^7}\) is crucial in calculating the probability of the circuit functioning correctly.
  • This expression represents the probability that all logic gates work without failure, where each gate operates independently with probability \(1-p\).
  • The exponential function is effective in modeling scenarios with huge numbers of trials or components, typical in fields like electronics and materials engineering.
Grasping how to apply such functions helps in solving complex probability problems.
Logarithms in Probability
Logarithms play a significant role in simplifying probability calculations, especially when dealing with exponential functions.
In the given problem, to find the gate failure probability, we take the natural logarithm of both sides of the equation \((1-p)^{10^7} = 0.95\).
  • This allows us to transform the exponential equation into a linear form: \(10^7 \ln(1-p) = \ln(0.95)\).
  • Logarithms thereby ease the calculation and solve for \(p\).
By understanding how to use logarithms, we can manipulate complex equations that arise in engineering tasks, facilitating a deeper comprehension of system reliability.

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