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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

Consider the time to recharge the flash in cell-phone cameras as in Example \(3-2 .\) Assume that the probability that a camera passes the test is 0.8 and the cameras perform independently. What is the smallest sample size needed so that the probability of at least one camera failing is at least \(95 \% ?\)

In \(1898,\) L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of \(0.61 .\) (a) What is the probability of more than one death in a corps in a year? (b) What is the probability of no deaths in a corps over five years?

A manufacturer of a consumer electronics product expects \(2 \%\) of units to fail during the warranty period. A sample of 500 independent units is tracked for warranty performance. (a) What is the probability that none fails during the warranty period? (b) What is the expected number of failures during the warranty period? (c) What is the probability that more than two units fail during the warranty period?

Let \(X\) denote the number of bits received in error in a digital communication channel, and assume that \(X\) is a binomial random variable with \(p=0.001\). If 1000 bits are transmitted, determine the following: (a) \(P(X=1)\) (b) \(P(X \geq 1)\) (c) \(P(X \leq 2)\) (d) mean and variance of \(X\)

A state runs a lottery in which six numbers are randomly selected from 40 without replacement. A player chooses six numbers before the state's sample is selected. (a) What is the probability that the six numbers chosen by a player match all six numbers in the state's sample? (b) What is the probability that five of the six numbers chosen by a player appear in the state's sample? (c) What is the probability that four of the six numbers chosen by a player appear in the state's sample? (d) If a player enters one lottery each week, what is the expected number of weeks until a player matches all six numbers in the state's sample?
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