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Chapter 3: Problem 26

The space shuttle flight control system called Primary Avionics Software Set (PASS) uses four independent computers working in parallel. At each critical step, the computers "vote" to determine the appropriate step. The probability that a computer will ask for a roll to the left when a roll to the right is appropriate is \(0.0001 .\) Let \(X\) denote the number of computers that vote for a left roll when a right roll is appropriate. What is the probability mass function of \(X ?\)

Short Answer

Expert verified
The PMF of \(X\) is given by the Binomial distribution with \(n=4, p=0.0001\).

Step by step solution

01

Understand the Problem

We are given a system with four independent computers. Each computer has a probability of 0.0001 of voting incorrectly (for a left roll instead of a right one). We denote this incorrect vote by the random variable \(X\). We need to find the probability mass function (PMF) of \(X\), the number of computers voting incorrectly.
02

Identify the Distribution

Since each computer votes independently and each vote is a binary outcome (correct or incorrect), \(X\) follows a Binomial Distribution. The number of trials \(n\) is 4 (one per computer), and the probability of success (incorrect vote) \(p\) is 0.0001.
03

Write the Binomial PMF Formula

The probability mass function for a Binomial random variable \(X\), with parameters \(n\) and \(p\), is given by \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \(k\) is the number of successes (incorrect votes) and \(\binom{n}{k}\) is the binomial coefficient.
04

Determine the PMF for This Scenario

Substitute \(n = 4\) and \(p = 0.0001\) into the PMF formula to find the probability for each possible value of \(k\) (0 to 4): - For \(k=0\), \(P(X=0) = \binom{4}{0} (0.0001)^0 (0.9999)^4\)- For \(k=1\), \(P(X=1) = \binom{4}{1} (0.0001)^1 (0.9999)^3\)- For \(k=2\), \(P(X=2) = \binom{4}{2} (0.0001)^2 (0.9999)^2\)- For \(k=3\), \(P(X=3) = \binom{4}{3} (0.0001)^3 (0.9999)^1\)- For \(k=4\), \(P(X=4) = \binom{4}{4} (0.0001)^4 (0.9999)^0\)

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Key Concepts

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

Binomial Distribution
The Binomial Distribution is a fundamental concept in probability theory. It models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.
In our context, it is used to analyze the voting behavior of computers in the space shuttle control system.Key features include:
  • A set number of trials (denoted by \(n\)). In the shuttle system example, \(n = 4\).
  • The probability of success in a single trial is constant, in this case \(p = 0.0001\) for an incorrect vote.
  • The outcome of each trial is binary, meaning each computer vote can be either correct or incorrect.
The formula for the Binomial probability mass function helps calculate the likelihood of a certain number of failures (incorrect votes) in four trials. This distribution is used to describe how individual probabilities contribute to the overall likelihood across different possible scenarios.
Random Variable
Random Variables are crucial in probability and statistics, representing numerical outcomes of random processes. In this exercise, the random variable \(X\) quantifies the count of computers voting incorrectly for a left roll.Key points to understand about random variables include:
  • They can take on different values, representing various outcomes related to the event of interest.
  • \(X\) takes discrete values from 0 to 4, indicating different numbers of incorrect votes.
  • This random variable is associated with probabilities, calculated through a probability mass function.
Understanding how to properly define and use random variables allows us to model and solve complex real-world problems, like maintaining the correct orientation of a space shuttle through calculations of probabilities.
Independent Trials
Independent Trials are the foundation of many probability models, including the Binomial Distribution. Conducting trials independently means that the outcome of one does not affect another. This concept is critical in our exercise. In the shuttle scenario:
  • Each computer votes independently, having no influence over another's decision.
  • Independence ensures that the total behavior is a simple combination of individual behaviors.
  • Implications of independence include utilizing the Binomial process, which assumes no inter-trial interference.
Such a system maintains reliability and ease of calculation, as probabilities can be straightforwardly multiplied without worrying about interdependencies.
Discrete Probability
Discrete Probability deals with events that have distinct outcomes, unlike continuous probability where outcomes can take any value in a range. It is crucial when dealing with random variables like in our example.Key aspects:
  • Discrete probability involves a countable number of outcomes. In our case, possible outcomes are \(X = 0, 1, 2, 3, \) and \(4\).
  • Probability mass function (PMF) is used to assign probabilities to each discrete outcome.
  • This gives insight into how likely each scenario is, allowing easy comparison and assessment.
Discrete probability is fundamental when you need precise calculations for distinct possibility levels, crucial for systems requiring high accuracy like space shuttle controls.

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Most popular questions from this chapter

An array of 30 LED bulbs is used in an automotive light. The probability that a bulb is defective is 0.001 and defective bulbs occur independently. Determine the following: (a) Probability that an automotive light has two or more defective bulbs. (b) Expected number of automotive lights to check to obtain one with two or more defective bulbs.

The probability that a visitor to a Web site provides contact data for additional information is \(0.01 .\) Assume that 1000 visitors to the site behave independently. Determine the following probabilities: (a) No visitor provides contact data. (b) Exactly 10 visitors provide contact data. (c) More than 3 visitors provide contact data.

Because all airline passengers do not show up for their reserved seat, an airline sells 125 tickets for a flight that holds only 120 passengers. The probability that a passenger does not show up is \(0.10,\) and the passengers behave independently. (a) What is the probability that every passenger who shows up can take the flight? (b) What is the probability that the flight departs with empty seats?

This exercise illustrates that poor quality can affect schedules and costs. A manufacturing process has 100 customer orders to fill. Each order requires one component part that is purchased from a supplier. However, typically, \(2 \%\) of the components are identified as defective, and the components can be assumed to be independent. (a) If the manufacturer stocks 100 components, what is the probability that the 100 orders can be filled without reordering components? (b) If the manufacturer stocks 102 components, what is the probability that the 100 orders can be filled without reordering components? (c) If the manufacturer stocks 105 components, what is the probability that the 100 orders can be filled without reordering components?

The number of telephone calls that arrive at a phone exchange is often modeled as a Poisson random variable. Assume that on the average there are 10 calls per hour. (a) What is the probability that there are exactly 5 calls in one hour? (b) What is the probability that there are 3 or fewer calls in one hour? (c) What is the probability that there are exactly 15 calls in two hours? (d) What is the probability that there are exactly 5 calls in 30 minutes?
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