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Q.4.12

Expert-verified

A First Course in Probability

Found in: Page 170

Book edition 9th

Author(s) Sheldon M. Ross

Pages 432 pages

ISBN 9780321794772

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Short Answer

There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?

The answer is $P (E)$ localid="1646910197664" $= \sum_{0 ⩽ m \leq (n + 1) / 2} (\begin{matrix} n + m - 1 \\ m \end{matrix}) p^{n - m} (1 - p)^{m}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1:Given Information

Let $X$ denotes the number of non functional components and let $E$ denotes the event. if no two nonfunctional components are to be constructive, then the space between the functional components must each contain at most one non functional components.

Step 2:Calculation

$P (E)$ $= \sum_{m = 0}^{n} P (E ∣ X = m) P (X = m)$ ,(From Bayes theorem)

$= \sum_{0 \leq m \leq n + 1) / 2} P (E ∣ X = m) P (X = m)$

localid="1646910154485" $= \sum_{0 \leq m \leq (n + 1) / 2} \frac{(\begin{matrix} n + m - 1 \\ m \end{matrix})}{(\begin{array}{l} n \\ m \end{array})} (\begin{array}{l} n \\ m \end{array}) p^{n - m} (1 - p)^{m}$

localid="1646910170371" $= \sum_{0 \leq m \leq (n + 1) / 2} (\begin{matrix} n + m - 1 \\ m \end{matrix}) p^{n - m} (1 - p)^{m}$

Step 3:Final Answer

The answer is $P (E)$ localid="1646910184882" $= \sum_{0 \leq m \leq (n + 1) / 2} (\begin{matrix} n + m - 1 \\ m \end{matrix}) p^{n - m} (1 - p)^{m}$

Most popular questions for Math Textbooks

Here is another way to obtain a set of recursive equations for determining $P_{n}$ , the probability that there is a string of $k$ consecutive heads in a sequence of $n$ flips of a fair coin that comes up heads with probability $p$ :

(a) Argue that for $k < n$ , there will be a string of $k$ consecutive heads if either

1. there is a string of $k$ consecutive heads within the first $n - 1$ flips, or

2. there is no string of $k$ consecutive heads within the first $n - k - 1$ flips, flip $n - k$ is a tail, and flips $n - k + 1, \dots, n$ are all heads.

(b) Using the preceding, relate $P_{n} t o P_{n - 1}$ . Starting with $P_{k} = p^{k}$ , the recursion can be used to obtain $P_{k + 1}$ , then $P_{k + 2}$ , and so on, up to $P_{n}$ .

Let $X$ be a Poisson random variable with parameter $λ$ . What value of $λ$ maximizes $P {X = k}, k \geq 0 ?$

Suppose that the distribution function of X given by

$F (b) = \{\begin{cases} 0 b < 0 \\ \frac{b}{4} 0 \leq b < 1 \\ \frac{1}{2} + \frac{b - 1}{4} 1 \leq b < 2 \\ \frac{11}{12} 2 \leq b < 3 \\ 1 3 \leq b \end{cases}$

(a) Find $P {X = i}, i = 1, 2, 3$ .

(b) Find $P \{\frac{1}{2} < X < \frac{3}{2}\}$ .

Suppose that it takes at least $9$ votes from a $12$ - member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is $. 2,$ whereas the probability that the juror votes an innocent person guilty is $. 1 .$ If each juror acts independently and if $65$ percent of the defendants are guilty, find the probability that the jury renders a correct decision. What percentage of defendants is convicted?

Let $X$ be a random variable having expected value $μ$ and variance $σ 2$ . Find the expected value and variance of $Y = \frac{X - μ}{σ}$ .

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