# A communication network between three locations, with probability of failure for each link

### Problem statement

Consider a communication network between three locations X, Y and Z. Assume there are links between each location and each link has a probability of failing as shown in the following table:

Link      Probability of Failure
XY             0.1
YZ             0.2
XZ             0.5


Let E be the event that location X is able to send a message to location Z (i.e., either via the link XZ or via Y using links XY and YZ). Determine P(E), the probability that the message has a route from X to Z by answering the following questions.

Deﬁne a sample space S consisting of 3-tuples representing the three links (XY; Y Z; XZ). If a link is working, assign it the value 1 and assign it 0 if the link is failing. List all outcomes in S.

### Progress

Trying to define the sample space here... So from x to z there are two options (xy,yz) and xz. For xz we can just draw 2 options a fail and a pass, but for xy we have to draw out 10 option with 9 passes and 1 fail and then 5(yz) for each of those options. Am I headed in the right direction?

• The sample space here is just all the 3-tuples from the set $\{0,1\}$. For example, define three random variables: $A,B,C$ (weird because you already have denoted something else in the problem with $X,Y,Z$). Let $A$ be the Random variable indicating the link condition between two points $X,Y$. It takes values $0,1$ with probabilities as given in the problem. The same way, define for $Y,Z$ and $Z,X$. Commented Dec 2, 2013 at 1:58
• I am lost could you eloaborate im totally new to this like a small example
– jai
Commented Dec 2, 2013 at 3:24

We have in the problem $3$ Random variables i.e. the variables that indicate whether the link between a pair of locations is working or not. Define the $3$ random variables as $A,B,C$. From the problem description, these random variables are Bernoulli (for definition of random variables and their types, I'm assuming you have some introduction to them) because they can take on $2$ values i.e. ${0,1}$. Let $0$ indicate a failure of link. Now the probability mass function $p$ for the random variable $A$, where $A$ is the random variable denoting the link condition between $X,Y$, is:$$p_A(A=a)=\left\{\begin{array}{} 0.1 & a=0 \\ 0.9 & a=1\end{array} \right.$$Likewise the other $2$ Random variables can also be defined. Now the task in hand is to first determine the sample space. Sample Space is the collection of all elementary outcomes of an experiment. It is composed of all possible individual events that can happen when the experiment is conducted. Here in our example, the experiment is just the presence or absence of links between the 3 points. We have $3$ random variables which indicate any such particular event. For example, $$\{A=0,B=1,C=0\}=\text{Event that link between XY is bad, YZ is good, XZ is bad}$$ I'll let you figure out all the possible events for this particular Sample space under consideration. The next task in hand is to figure out the probability that $X$ can speak to $Z$ either directly or indirectly. The following need to be done to figure this out:
1. Look at the sample space; see all the events that can lead to a path for transmission from $X$ to $Z$.
2. Find the probability of each of them happening separately, an example calculation is as follows: (assuming that the links work independently of each other)$$\mathbb{P}[\{A=0,B=1,C=0\}]=\mathbb{P}[\{A=0\}]\mathbb{P}[\{B=1\}]\mathbb{P}[\{C=0\}]$$ where $\mathbb{P}$ stands for probability. The above is due to independence of random variables i.e. the link condition between $X,Y$ is irrelevant to the link condition between $Y,Z$ and also to $Z,X$. Simply speaking, the links fail on their own randomly without other links influencing them.
3. Add all the probabilites that you found in step $2$. This step is valid because the events you're considering are disjoint. At any time, the link conditions take on only one value. The event $\mathbf{E}$ in question is composed of multiple disjoint events and therefore, $$\mathbb{P}[\mathbf{E}]=\mathbb{P}[\{E_1\cup E_2,\dots,\cup E_n\}]=\mathbb{P}(\{E_1\}]+\mathbb{P}(\{E_2\}]+\dots+\mathbb{P}(\{E_n\}]$$ where $E_i$'s indicate the different events that can lead to a transmission between $X$ and $Z$.
1. Let $A,B,C$ stand as $3$ letters denoting the link condition between the pairwise communication points.
2. $A$ is assigned a $0$ if the link is bad or a $1$ if the link is good. Now form the following table for each of the letters and the values that they can take with their corresponding probabilities. $$\begin{array}{c|c}A & \text{Probability that A takes on the value} \\ 0 & 0.1 \\ 1 & 0.9\end{array}$$ Similarly, form the tables for $B,C$.
3. The sample space, denoted by $\Omega$, is just all the possible combinations of the values of $A,B,C$. $$\Omega=\{\{000\},\{001\},\dots,\{111\}\}$$
4. The rest of the question is straight forward as explained in the first $3$ initial steps.