1  Exercises 1

Counting (Combinatorics)

Exercise 1.1 Separated questions:

  1. In computing, the binary system is used to code characters. A bit (binary digit) is an element that takes the value 0 or the value 1. With 8 binary digits (one byte), how many characters can we code?
  1. In a sports competition grouping 18 athletes, one gold, one silver and one bronze medal are awarded. How many distributions are there possible (before the competition, of course…)?
  1. A sports tournament has 8 participating teams. Each team must meet all the others once. How many games must we organize?

Exercise 1.2 Five cards are drawn simultaneously from a deck of 52 cards.

  1. How many different draws can we get?

  2. How many different draws can we get? containing:

    1. 5 diamonds or 5 hearts;

    2. 2 hearts and 3 spades;

    3. at least 1 king

    4. at most 1 king.

Events

Exercise 1.3 Let \(A\),\(B\) and \(C\) be three events of a measurable space \((\Omega,\mathcal{A})\). Express in terms of \(A\),\(B\) and \(C\) and the operations (Union, intersection and complementary) the following events below:

  1. \(A\) alone (among the 3 events) occurs.

  2. \(A\) and \(C\) occur, but not \(B\).

  3. All three events occur.

  4. At least one of the 3 events occurs.

  5. None of the three events occur.

Probability

Exercise 1.4 Let the events \(A\) and \(B\) such that \(P(A) = x, P(B) = y\) and \(P(A \cup B) = z\). Find the following probabilities: \(P(A \cap B), P(\bar{A} \cup \bar{B})\), and \(P(A \cap \bar{B})\).

Exercise 1.5 An urn contains 13 balls, 6 black, 3 white and 4 red. We draw 4 balls. Let

  • E: ” get exactly 2 white “.
  • F: ” get exactly 2 red “.

We suppose that we draw without replacement. Calculate the following probabilities: \(P(E \cap F), P_F (E), P_E(F)\). Are the events E and F independent?

Discrete random variables

Exercise 1.6 Two balls are chosen randomly from an urn containing 8 white balls, 4 black balls and 2 orange balls. Assume we receive 2 euros for each black ball drawn and we lose 1 euro for each white ball drawn. Let’s refer to the net gains by \(X\).

  1. What are the possible values for \(X\) and the probabilities associated with these values ?

  2. What is the expected value of \(X\)?

Exercise 1.7 Let \(X\) be a random variable that follows the uniform distribution (e.g. equiprobability of values of \(X\)) on the set \(X(\Omega) = \{-3, -2, 1, 4\}\).

  1. Give the law of \(X\).

  2. Calculate \(E(X)\) and \(V(X)\).

    We define the random variable \(Y=(X+1)^2\).

  3. Give \(Y(\Omega)\) and the law of \(Y\).

  4. Calculate \(E(Y)\) using two different methods.

Exercise 1.8 Let X and Y be discrete random variables whose joined distribution is given by the following table:

\(X\)\\(Y\) -1 0 2 5
0 0.10 0.05 0.15 0.05
1 0.15 0.20 0.25 0.05

  1. What is the marginal distribution of X?

  2. What is the marginal distribution of Y?

  3. Calculate \(P(Y \geq 0 / X = 1)\).

  4. Compute \(E(X)\), \(E(Y)\), and \(cov(X,Y)\).

  5. Are the variables \(X\) and \(Y\) independent?

Exercise 1.9 We roll a six-sided die. We assume that the probability of each face appearing is proportional to the number written on it.
Let \(X\) denote the number of points obtained.

  1. Give the distribution (law) of \(X\).
  2. Recall the definitions of \(E(X)\), \(\mathrm{Var}(X)\), and \(\sigma(X)\), and compute their values.
  3. Provide a graphical representation of the cumulative distribution function (CDF) of the variable \(X\).
  4. Compute the probability of obtaining an even number, then that of obtaining an odd number.

Exercise 1.10 We consider the situation where two players each roll a fair die in turn.
Let \(Z_1\) (respectively \(Z_2\)) denote the result of player 1 (resp. player 2), assuming the two rolls are independent.
We define \(A\) as the average of the two results and \(B\) as their maximum, that is:

\[ A = \frac{Z_1 + Z_2}{2}, \qquad B = \max(Z_1, Z_2). \]

  1. What is the joint distribution (law) of the pair \((Z_1, Z_2)\)?
  2. Compute the expectation and the variance of \(A\).
  3. What is the distribution of \(B\)? Compute its expectation, variance, and standard deviation.
  4. Are the variables \(A\) and \(B\) independent?

Exercise 1.11 An urn contains 2 balls with number 20, 4 balls with number 10 and 4 balls with number 5.

  1. An experiment consists in drawing simultaneously 3 balls from the urn. Calculate probability p so that the sum of the numbers drawn is equal to 30.

  2. This experiment is repeated 4 times, each time putting the three balls drawn back into the urn. Let \(X\) be the random variable indicating the number of draws giving a sum of numbers equal to 30.

    1. What is the probability law of \(X\). Give its expected and standard deviation.

    2. Calculate the probability of having at least once the sum 30 out of 4 draws.

Exercise 1.12 You need someone to help you move. When you call a friend, there’s a one out of four chances he will accept. Let \(X\) be the random variable that represents the number of friends you will need to contact to get this help.

  1. Determine the probability distribution of \(X\).

  2. Calculate \(P(X\leq 3)\).

  3. Calculate \(E(X)\).

Exercise 1.13 The number of power outages that occur in a certain region over a one-year timeframe follows a Poisson distribution of parameter \(\lambda=3\).

  1. Calculate the probability that within a period of one year one failure occurs exactly.

  2. Assuming there is an independence of outages from one year to another, calculate the probability that in the upcoming ten years there will be at least one year in which exactly one failure occurs.