Successful Probability of a Communication Network Diagram

Problem 742

Consider the network diagram in the figure. The diagram consists of five links and each of them fails to communicate with probability $p$. Answer the following questions about this network.

(1) Determine the probability that there exists at least one path from A to B where every link on the path functions without errors. Express the answer in term of $p$.

(2) Assume that exactly one link has failed. In this case, what is the probability that there is a successful path from A to B, that is, each link on the path has not fail.

If there is path along which no link has failed, then Link 5 must be successful and at least one of the upper subpath (Link 1 and Link 3) or the lower subpath (Link 2 and Link 4) should not fail.
Thus the desired probability is
\begin{align*}
& P(\text{there is a successful path})\\
&= P(\text{the upper or lower subpath is successful}) * P(\text{Link 5 is successful}).
\end{align*}

Furthermore, we have using the property of probability that
\begin{align*}
&P(\text{the upper or lower subpath is successful}) \\
&= 1 – P(\text{both upper and lower subpaths failed})\\
&= 1 – P(\text{upper subpaths failed}) * P(\text{lower subpaths failed})
\end{align*}

Now, the upper subpaths fail if Link 1 or Link 3 fails. That is,
\begin{align*}
&P(\text{upper subpaths failed}) \\
&= 1 – P(\text{Link 1 is successful}) * P(\text{Link 3 is successful})\\
&= 1 – (1 – p) ^2\\
\end{align*}
Similarly, we get $P(\text{lower subpaths failed}) = 1 – (1-p)^2$.
Combining these results, we obtain the desired probability
\begin{align*}
& P(\text{there is a successful path})\\
&= \left(1 – \left[ 1 – (1-p)^2 \right]^2 \right) * (1-p)
\end{align*}

Solution of (2)

There is a successful path only when Link 5 has not failed.

Since all five links have the same failing probability, the probability that Link 5 has failed is $1/5$. Thus, with probability $1 – 1/5 = 4/5$, there is a successful path from A to B.

Probability Problems about Two Dice
Two fair and distinguishable six-sided dice are rolled.
(1) What is the probability that the sum of the upturned faces will equal $5$?
(2) What is the probability that the outcome of the second die is strictly greater than the first die?
Solution.
The sample space $S$ is […]

Probability that Alice Tossed a Coin Three Times If Alice and Bob Tossed Totally 7 Times
Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$.
Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three […]

Upper Bound of the Variance When a Random Variable is Bounded
Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$.
\[V(X) \leq \frac{c^2}{4}.\]
Proof.
Recall that […]

Lower and Upper Bounds of the Probability of the Intersection of Two Events
Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that
\[a \leq P(A \cap B) \leq b\]
and $P(A \cap B)$ could […]

How to Prove Markov’s Inequality and Chebyshev’s Inequality
(a) Let $X$ be a random variable that takes only non-negative values. Prove that for any $a > 0$,
\[P(X \geq a) \leq \frac{E[X]}{a}.\]
This inequality is called Markov's inequality.
(b) Let $X$ be a random variable with finite mean $\mu$ and variance $\sigma^2$. Prove that […]

What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?
Four fair coins are tossed.
(1) What is the probability that all coins land heads?
(2) What is the probability that all coins land heads if the first coin is heads?
(3) What is the probability that all coins land heads if at least one coin lands […]

Probability of Getting Two Red Balls From the Chosen Box
There are two boxes containing red and blue balls. Let us call the boxes Box A and Box B. Each box contains the same number of red and blue balls. More specifically, Box A has 5 red balls and 5 blue balls. Box B has 20 red balls and 20 blue balls. You choose one box. Then draw two […]

Probabilities of An Infinite Sequence of Die Rolling
Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability.
(1) At least one die lands on the face 5 in the first $n$ rolls.
(2) Exactly $k$ dice land on the face 5 […]