To access a company's private network from outside, employee connections are randomly routed through three different remote servers, denoted $\mathrm{A}, \mathrm{B}$ and C. These servers have different technical characteristics and connections are distributed as follows:
- $25\%$ of connections are routed through server A;
- $15\%$ of connections are routed through server B;
- the remaining connections are made through server C.
A connection is said to be stable if the user does not experience a disconnection after authentication to the servers. The IT maintenance team has statistically observed that, under normal server operation:
- $90\%$ of connections via server A are stable;
- $80\%$ of connections via server B are stable;
- $85\%$ of connections via server C are stable.
Part AWe are interested in the state of a connection made by an employee of the company. We consider the following events:
- A: ``The connection was made via server A'';
- B: ``The connection was made via server B'';
- C: ``The connection was made via server C'';
- S: ``The connection is stable''.
We denote by $\bar{S}$ the complementary event of event $S$.
- Copy and complete the weighted tree below modelling the situation described in the problem.
- Prove that the probability that the connection is stable and passes through server B is equal to 0.12.
- Calculate the probability $P(C \cap \bar{S})$ and interpret the result in the context of the exercise.
- Prove that the probability of event $S$ is $P(S) = 0.855$.
- Now suppose that the connection is stable. Calculate the probability that the connection was made from server B. Give the answer rounded to the nearest thousandth.
Part BAccording to Part A, the probability that a connection is unstable is equal to 0.145.
- In order to detect server malfunctions, we study a sample of 50 connections to the network, these connections being chosen at random. We assume that the number of connections is large enough that this choice can be treated as sampling with replacement.
Let $X$ denote the random variable equal to the number of unstable connections to the company's network, in this sample of 50 connections. a. We admit that the random variable $X$ follows a binomial distribution. Specify its parameters. b. Give the probability that at most eight connections are unstable. Give the answer rounded to the nearest thousandth. - In this question, we now form a sample of $n$ connections, still under the same conditions, where $n$ denotes a strictly positive natural number. We denote by $X_n$ the random variable equal to the number of unstable connections and we admit that $X_n$ follows a binomial distribution with parameters $n$ and 0.145. a. Give the expression as a function of $n$ of the probability $p_n$ that at least one connection in this sample is unstable. b. Determine, by justifying, the smallest value of the natural number $n$ such that the probability $p_n$ is greater than or equal to 0.99.
- We are interested in the random variable $F_n$ equal to the frequency of unstable connections in a sample of $n$ connections, where $n$ denotes a strictly positive natural number. We thus have $F_n = \frac{X_n}{n}$, where $X_n$ is the random variable defined in question 2. a. Calculate the expectation $E(F_n)$. We admit that $V(F_n) = \frac{0.123975}{n}$. b. Verify that: $P\left(\left|F_n - 0.145\right| \geqslant 0.1\right) \leqslant \frac{12.5}{n}$ c. A company manager studies a sample of 1000 connections and observes that for this sample $F_{1000} = 0.3$. He suspects a server malfunction. Is he right?