Q. 2.5 - A First Course in Probability

Question

A system is composed of $5$ components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector $(x_1, x_2, x_3, x_4, x_5)$, where $x_i$ is equal to $1$ if component $i$ is working and is equal to $0$ if component $i$ is failed.

(a) How many outcomes are in the sample space of this experiment?

(b) Suppose that the system will work if components $1$ and $2$ are both working, or if components $3$ and $4$ are both working, or if components $1$, $3$, and $5$ are all working. Let W be the event that the system will work. Specify all the outcomes in W.

(c) Let $A$ be the event that components $4$ and $5$ are both failed. How many outcomes are contained in the event $A$?

(d) Write out all the outcomes in the event $AW$.

Step-by-Step Solution

Answer

(a) Total no. of outcomes are $32$.

(b) All the outcomes in W are $$\begin{gathered} \left\{\left(0,0,1,1,0\right), \left(0,0,1,1,1\right), \left(0,1,1,1,0\right), \left(0,1,1,1,1\right), \left(1,0,1,0,1\right), \\ \left(1,0,1,1,0\right), \left(1,0,1,1,1\right), \left(1,1,0,0,0\right), \left(1,1,0,0,1\right), \left(1,1,0,1,0\right), \\ \left(1,1,0,1,1\right), \left(1,1,1,0,0\right), \left(1,1,1,0,1\right), \left(1,1,1,1,0\right), \left(1,1,1,1,1\right)\right\} \end{gathered}$$

(c) No. of outcomes in the event A are $8$.

(d) All the outcomes in the event AW are $\left\{\left(1,1,0,0,0\right),\left(1,1,1,0,0\right)\right\}$

Part (a) Step 1. Find the total no. of outcomes.

Every component can be in either of one state - working or failed. So, all the five components can be either $1$ or $0$, that is two state.

Therefore, the total no. of outcomes are $=2^5=32$.

Part (b) Step 1. List all the outcomes in W .

The system will work if components $1$ and $2$ are working, so the outcomes are:

$$\begin{gathered} \left(1,1,0,0,0\right), \left(1,1,0,0,1\right), \left(1,1,0,1,0\right), \left(1,1,0,1,1\right), \left(1,1,1,0,0\right), \\ \left(1,1,1,0,1\right), \left(1,1,1,1,0\right), \left(1,1,1,1,1\right) \end{gathered}$$

The system will work if components $3$ and $4$ are working, so the outcomes are: 

$$\begin{gathered} \left(0,0,1,1,0\right), \left(0,0,1,1,1\right), \left( 0,1,1,1,0\right), \left(0,1,1,1,1\right), \left(1,0,1,1,0\right), \\ \left(1,0,1,1,1\right), \left(1,1,1,1,0\right), \left(1,1,1,1,1\right) \end{gathered}$$

The system will work if components $1,3, \text{and} 5$ are working, so the outcomes are: 

$\left(1,0,1,0,1\right), \left(1,0,1,1,1\right), \left(1,1,1,0,1\right), \left(1,1,1,1,1\right)$

Therefore, the outcomes in $W$ are

$$\begin{gathered} \left\{\left(0,0,1,1,0\right), \left(0,0,1,1,1\right), \left(0,1,1,1,0\right), \left(0,1,1,1,1\right), \left(1,0,1,0,1\right), \\ \left(1,0,1,1,0\right), \left(1,0,1,1,1\right), \left(1,1,0,0,0\right), \left(1,1,0,0,1\right), \left(1,1,0,1,0\right), \\ \left(1,1,0,1,1\right), \left(1,1,1,0,0\right), \left(1,1,1,0,1\right), \left(1,1,1,1,0\right), \left(1,1,1,1,1\right)\right\} \end{gathered}$$

Part (c) Step 1. Find the no. of outcomes in A .

$A$ represents the event that components $4$ and $5$ both failed. So the outcomes are:

$$\begin{gathered} \left\{\left(0,0,0,0,0\right), \left(1,0,0,0,0\right), \left(0,1,0,0,0\right), \left(0,0,1,0,0\right), \\ \left(1,1,0,0,0\right), \left(0,1,1,0,0\right), \left(1,1,1,0,0\right), \left(1,0,1,0,0\right)\right\} \end{gathered}$$

Therefore, the no. of outcomes in the event $A$ are $8$.

Part (d) Step 1. List the outcomes in A W .

$$\begin{gathered} A=\left\{\left(0,0,0,0,0\right), \left(1,0,0,0,0\right), \left(0,1,0,0,0\right), \left(0,0,1,0,0\right), \\ \left(1,1,0,0,0\right), \left(0,1,1,0,0\right), \left(1,1,1,0,0\right), \left(1,0,1,0,0\right)\right\} \end{gathered}$$

$$\begin{gathered} W=\left\{\left(0,0,1,1,0\right), \left(0,0,1,1,1\right), \left(0,1,1,1,0\right), \left(0,1,1,1,1\right), \left(1,0,1,0,1\right), \\ \left(1,0,1,1,0\right), \left(1,0,1,1,1\right), \left(1,1,0,0,0\right), \left(1,1,0,0,1\right), \left(1,1,0,1,0\right), \\ \left(1,1,0,1,1\right), \left(1,1,1,0,0\right), \left(1,1,1,0,1\right), \left(1,1,1,1,0\right), \left(1,1,1,1,1\right)\right\} \end{gathered}$$

Therefore, the outcomes in $AW$ are $\left\{\left(1,1,0,0,0\right),\left(1,1,1,0,0\right)\right\}$