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Chapter 2: Problem 163

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 \( p \) is approximately \( 5.129 \times 10^{-9} \).

Step by step solution

01

Define the Problem

We have 10 million gates, and we need the circuit to function with 95% probability. Each gate has a failure probability of \( p \), and if any gate fails, the entire circuit fails. We aim to find \( p \).
02

Calculate Probability of a Single Logic Gate Working

If the probability of a gate failing is \( p \), then the probability of a gate working properly is \( 1 - p \).
03

Understand the Circuit Functionality Probability

The circuit functions if all 10 million gates function properly. Since the failures are independent, the probability of all gates working is \((1-p)^{10^7}\).
04

Set Up the Equation for Circuit Functioning

Since the probability of the circuit functioning should be 0.95, we have the equation: \((1-p)^{10^7} = 0.95\).
05

Solve for \( p \)

To solve \((1-p)^{10^7} = 0.95\), take the natural logarithm of both sides: \[ \ln((1-p)^{10^7}) = \ln(0.95) \]This simplifies to:\[ 10^7 \ln(1-p) = \ln(0.95) \]Solving for \( p \), we get:\[ \ln(1-p) = \frac{\ln(0.95)}{10^7} \]You can use approximation \( \ln(1-x) \approx -x \) for small \( x \), leading to:\[ -p \approx \frac{\ln(0.95)}{10^7} \]Calculate \( p \):\[ p \approx -\frac{\ln(0.95)}{10^7} \]
06

Compute the Value of \( p \)

Calculate the value:\[ \ln(0.95) \approx -0.05129 \]Substitute the value in the equation:\[ p \approx -\frac{-0.05129}{10^7} \approx 5.129 \times 10^{-9} \]
07

Final Conclusion

The probability of a single gate failing should be approximately \( 5.129 \times 10^{-9} \) to ensure that the circuit functions with a 95% probability.

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Key Concepts

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

Independent Events
In probability theory, independent events are those whose occurrence or outcome does not affect one another. This means the probability of several events occurring together is simply the product of their individual probabilities.

For example, when considering an integrated circuit with multiple gates, if one gate fails, it does not change the probability of another gate failing. Each gate operates independently of the others.
  • The failure or success of each gate is not influenced by other gates.
  • We calculate the overall probability by multiplying the individual probabilities of each gate functioning properly.
Understanding this independence is crucial when analyzing systems like circuits where components work autonomously.
Failure Probability
Failure probability refers to the likelihood that a component or system will fail to perform as intended. In our exercise with logic gates, we are given that each gate has its own probability of failing, denoted as \( p \).

Since the circuit fails if any gate fails, we must focus on minimizing \( p \) to ensure overall reliability.
  • A lower value of \( p \) suggests a higher chance of the circuit functioning correctly.
  • We need a precise calculation of \( p \) for designing reliable circuits.
By setting the probability of the whole circuit working to 0.95, we calculate how small \( p \) must be to meet this requirement. This links directly to circuit reliability.
Logarithm Approximation
Logarithm approximation is a handy tool for dealing with exponential equations, especially when computing manual calculations. In particular, the natural logarithm approximation \( \ln(1-x) \approx -x \) for small \( x \) is often used.

In the context of our problem, where \( 1-p \approx 1 \), this approximation helps simplify the equation \( (1-p)^{10^7} = 0.95 \).
  • Using the approximation makes solving complex exponential equations both feasible and convenient.
  • It only works well when \( x \) is very small, which is typical in reliability and failure probability problems.
This simplification helps us find an easily calculable estimate for the failure probability \( p \).
Circuit Reliability
Circuit reliability is the likelihood that an entire system, such as our integrated circuit, will function without failure over a specified period. It's critical in electronics and systems design where operational success hinges on various interdependent elements functioning as intended.

Achieving high reliability requires understanding and controlling individual component failure rates, like the logic gates in our exercise.
  • For our circuit to be reliable, each gate must operate correctly, making their probabilities crucial metrics.
  • Designers aim for minimal failure probabilities to ensure overall system durability.
By determining appropriate probabilities for each component, systems can be designed to function successively, providing assurance and confidence in their performance.

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Most popular questions from this chapter

It is known that two defective copies of a commercial software program were erroneously sent to a shipping lot that now has a total of 75 copies of the program. A sample of copies will be selected from the lot without replacement. (a) If three copies of the software are inspected, determine the probability that exactly one of the defective copies will be found. (b) If three copies of the software are inspected, determine the probability that both defective copies will be found. (c) If 73 copies are inspected, determine the probability that both copies will be found. (Hint: Work with the copies that remain in the lot.)

If the last digit of a weight measurement is equally likely to be any of the digits 0 through 9 (a) What is the probability that the last digit is \(0 ?\) (b) What is the probability that the last digit is greater than or equal to \(5 ?\)

A computer system uses passwords that contain exactly eight characters, and each character is one of 26 lowercase letters \((a-z)\) or 26 uppercase letters \((A-Z)\) or 10 integers \((0-9)\). Let \(\Omega\) denote the set of all possible passwords, and let \(A\) and \(B\) denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in \(\Omega\) are equally likely. Determine the probability of each of the following: (a) A (b) \(B\) (c) A password contains at least 1 integer. (d) A password contains exactly 2 integers.

The sample space of a random experiment is \(\\{a, b,\) \(c, d, e\\}\) with probabilities \(0.1,0.1,0.2,0.4,\) and \(0.2,\) respectively. Let \(A\) denote the event \(\\{a, b, c\\},\) and let \(B\) denote the event \(\\{c, d, e\\} .\) Determine the following: (a) \(P(A)\) (b) \(P(B)\) (c) \(P\left(A^{\prime}\right)\) (d) \(P(A \cup B)\) (e) \(P(A \cap B)\)

In the manufacturing of a chemical adhesive, \(3 \%\) of all batches have raw materials from two different lots. This occurs when holding tanks are replenished and the remaining portion of a lot is insufficient to fill the tanks. Only \(5 \%\) of batches with material from a single lot require reprocessing. However, the viscosity of batches consisting of two or more lots of material is more difficult to control, and \(40 \%\) of such batches require additional processing to achieve the required viscosity. Let \(A\) denote the event that a batch is formed from two dif- ferent lots, and let \(B\) denote the event that a lot requires additional processing. Determine the following probabilities: (a) \(P(A)\) (b) \(P\left(A^{\prime}\right)\) (c) \(P(B \mid A)\) (d) \(P\left(B \mid A^{\prime}\right)\) (e) \(P(A \cap B)\) (f) \(P\left(A \cap B^{\prime}\right)\) (g) \(P(B)\)
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