semester, Test Probabilities length: 2 hours – 20 points, inner coefficient 1.5

2019-2021 – S2 – Mathematics – TEST Probabilities – page 1 / 3

IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2019-2021 25/05/2020

MATHEMATICS – 2

semester, Test Probabilities length: 2 hours – 20 points, inner coefficient 1.5

This document is a statement of your last test of the second semester. It was provided to you just before 1 p.m. and is to be processed at home, in draft form. You must enter your answers on ExoOnLine in the six tests provided (one per exercise). This is to be done in two hours, but for possible

embarrassment related to the entry, you are given an extra 15 minutes: you have until 3:15 pm to have finished entering your answers online (students benefiting from a third time will have until 4 pm).

Exercise 1: MCQ (2 points) – tick your answers below

1 good answer for each question; wrong, multiple or missing answer: 0 point 1) Which one displays the correct order?

C_n^p ≤P_n^p ≤n^p n^p ≤P_n^p ≤C_n^p n^p ≤C_n^p ≤P_n^p C_n^p ≤n^p ≤P_n^p 2) Rolling several dice at the same time inevitably leads to:

permutations p-lists combinations it depends

3) If A and B each have a 50% chance of occurring, then the chances of A∩B are:

0% 25% 50% it depends

4) On a choice tree, which of the following probabilities can be read directly?

events and conditional conditional and the three

intersections and events intersections previous kinds

Exercise 2: (2 points) Sets and cardinal numbers

By adapting the formula ^{Card A}

(

)

( )

( )

(

)

( )

Card A∪ ∪B C , thus a formula using only simple sets and intersections.

For the convenience of writing on a computer, you will use lowercase n for the intersection and lowercase u for the union, for example: Card(AnB), Card(AuB).

Exercise 3: (6 points) Combinatorics

1) How many integers, between 10,000 and 99,999, contain neither 0, 1, nor 2? 0.75 pt

2520 7776 15625 16807

2) How many integers, between 0 and 99,999, contain neither 0, 1, nor 2? 1 pt

3619 9330 19607 20515

3) Consumers are asked to choose their four favourite products from a list of twelve products, listing the four in order of preference. How many different rankings are possible? 1 pt 4) There are 15 stores on one street, 5 of which are clothing stores and 3 of which are grocery stores. The

town hall decides to award a bonus to three shops whose names will be drawn at random from among the 15.

a. How many different draws of three stores are possible? 0.75 pt

b. Of these possible draws, how many include two clothing stores and one grocery store? 1.25 pt c. Which is more likely: that the future draw has 1 clothing store or 2 clothing stores? (your answer will of

course have to be justified by calculations) 1.25 pt

2019-2021 – S2 – Mathematics – TEST Probabilities – page 2 / 3 Exercise 4: (2 points) Conditional probabilities

A random experiment consists of rolling a die and noting its result.

Give an example of two compatible (= not mutually exclusive) but independent events (and justify).

Exercise 5: (3 points) Conditional probabilities

10% of French people over the age of 18 are left-handed (event "L"). 70% of them have passed their baccalaureate with honours ("H" event), compared with 40% of right-handed people.

1) A student has just seen his baccalaureate results and exclaims, "I got it with honours! ". What is the

probability that this student is left-handed? 2 pts

2) Are the events L and H independent? 1 pt

Exercise 5: (5 points) Simple probabilities and distributions Let's take again an example seen in tutorial: the roll of two dice at the end of which we note the total of the two dice. Remember that the 36 possible pairs {(1,1), (1,2), ..., (3,3), (3,4), (3,5), ..., (5,6), (6,6)} are equally likely to occur and that, for example, the probability of making a total of 9 is p(9) = 4/36, because 4 pairs (out of the 36) give a total of 9: (3,6), (4,5), (5,4), (6,3).

We note A the event "obtain a total strictly lower than 8", B the event "obtain a total of 8 or 9" and C the event "obtain a total strictly higher than 9".

1) a. Give the probabilities of the events A, B and C. 0.75 pt

b. Do these three events form a partition of the sample space? 0.75 pt 2) When a game is played, getting event A gives the player no winnings; event B gives him a €1 win and

event C a €3 win.

a. Give the probability distribution of the random variable X : « gain of one game ». 0.5 pt b. To play a game, a player must bet one euro. On a large number of games, who wins? The organizer or

the players? 1 pt

c. The game organizer expects a total of 30,000 games to be played per month, once the game has been online for some time. Give a 95% confidence interval of the monthly income that the organizer can

predict. 2 pts

____________________ TEST END ____________________

2019-2021 – S2 – Mathematics – TEST Probabilities – page 3 / 3

IUT TC MATHEMATICS FORM "COMBINATORICS AND PROBABILITIES"