Question

Assignment 3(Chapter 3) To be submitted due to Dec.3 1. Each time a component is tested, the trial is a success (S) or failure (F). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let Y denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of Y, and state which Y value is associated with each one.

Answer 1

3. Here X denotes the number of lines in use at a specified time. Therefore, the probability P(X = x) = p(x).

a. P(at most three lines are in use)

$\begin{gathered} = P(X \leqslant 3) \hfill \\ = p(0) + p(1) + p(2) + p(3) \hfill \\ = 0.1 + 0.15 + 0.2 + 0.25 \hfill \\ = 0.7 \hfill \\ \end{gathered}$

b. P(fewer than three lines are in use)

$\begin{gathered} = P(X < 3) \hfill \\ = p(0) &plus; p(1) &plus; p(2) \hfill \\ = 0.1 &plus; 0.15 &plus; 0.2 \hfill \\ = 0.45 \hfill \\ \end{gathered}$

c. P(at least three lines are in use)

$\begin{gathered} = P(X \geqslant 3) \hfill \\ = p(3) &plus; p(4) &plus; p(5) &plus; p(6) \hfill \\ = 0.25 &plus; 0.2 &plus; 0.06 &plus; 0.04 \hfill \\ = 0.55 \hfill \\ \end{gathered}$

d. P(between two and five lines, inclusive, are in use)

$\begin{gathered} = P(2 \leqslant X \leqslant 5) \hfill \\ = p(2) &plus; p(3) &plus; p(4) &plus; p(5) \hfill \\ = 0.2 &plus; 0.25 &plus; 0.2 &plus; 0.06 \hfill \\ = 0.71 \hfill \\ \end{gathered}$

e. P(between two and four lines, inclusive, are not in use) = 1 - P(between two and four lines, inclusive, are in use)

$\begin{gathered} = 1 - P(2 \leqslant X \leqslant 4) \hfill \\ = 1 - [p(2) &plus; p(3) &plus; p(4)] \hfill \\ = 1 - [0.2 &plus; 0.25 &plus; 0.2] \hfill \\ = 1 - 0.65 \hfill \\ = 0.35 \hfill \\ \end{gathered}$

f. P(at least four lines are not in use) = 1 - P(at least four lines are in use)

$\begin{gathered} = 1 - P(X \geqslant 4) \hfill \\ = 1 - [p(4) &plus; p(5) &plus; p(6)] \hfill \\ = 1 - [0.2 &plus; 0.06 &plus; 0.04] \hfill \\ = 1 - 0.3 \hfill \\ = 0.70 \hfill \\ \end{gathered}$