IUT of Saint-Etienne – Sales and Marketing department

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IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2016-2018 05/2017

MATHEMATICS – 2

^nd

semester, Test 2 length : 2 hours – coefficient 1/2

Graphic calculators are allowed. Any personal sheet is forbidden.

The presentation and the quality of your writings will be taken into account.

Your rounded results must be expressed with at least four significant figures.

Exercise 1 : MCQ (2 points) – tick the right boxes below

One correct answer only per question - 0 point in case of wrong/missing/multiple answer at a question 1)

(

^A^∩^B

) (

^∪ ^A^∩^B

)

^...

⊂A ⊃^B ⊃ ∩A B ⊂ ∪^A ^B

2) If ^Card

(

^A^{∩ =}^B

)

^Card

( )

^A ⁻^Card

( )

^B ^{, then:}

A = B ^Card

(

^A^{∪ = ×}^B

)

² ^Card

( )

^B ^Card

( )

^A ⁼^Card

( )

^B A and B intersect 3) If two events A and B are independent, then:

p(A) + p(B) = 1 pA(B) = pB(A) p(A∩B) = p(B) p (B) = p (B) _A _A 4) Which one is the correct inequality?

P C

p p p

n n

n ≤ ≤ n^p ≤C_n^p ≤P_n^p C_n^p ≤P_n^p ≤n^p C_n^p ≤n^p ≤P_n^p

Exercise 2 : Cardinal numbers (4 points)

A survey consists in analysing the sales quantities of two products a and b in a shop. Within 200 clients, 57 bought the objet a, 103 bought the objet b, 38 bought both objects. We name A the set of clients who bought the object a and B the set of clients who bought the object b.

1) Calculate ^Card

(

^A^∪^B

)

and then give a concrete meaning of this number. 1 pt

2) Build a contingency table for A, B and their contraries. 1 pt

Full name : Group : B1

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3) Thanks to this table, say (justify your answers and name the corresponding sets):

a. How many people bought neither a nor b. 0.5 pt

b. How many people bought exactly one of both objects. 0.5 pt

4) What is the probability that a client bought the object b, given that this person bought the object a.

1 pt

Exercise 3 : Counting (2 points)

Seven candidates (A, B, C, D, E, F and G) stand in municipal elections. You will answer the following questions by justifying the counting tools used.

1) At the count of the first round, these candidates are ranked in descending order of the number of votes obtained. How many different rankings are possible for these seven candidates? 1 pt

2) Both candidates who scored the best are qualified for the second round. How many different second

rounds are possible? 1 pt

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Exercise 4 : Counting and probabilities (5 points)

A magician shows a set of 32 playing cards (4 colours, 8 cards per colour: hearts, spades, clubs, diamonds) to the audience who only sees the reverse side.

1) First game: the magician asks person A to pick a card at random, memorise it, and then put it back in the set he immediately mixes. He does the same with three other people B, C and D.

a. How many different draws, in the order A, B, C, D, are possible? 0.5 pt

b. What is the probability that A would pick a heart? 0.5 pt

c. What is the probability that A would pick a heart and B would pick a spade? 0.5 pt

2) Second game: the magician asks person A to make a simultaneous draw of four cards.

a. Calculate the number of different possible draws. 0.5 pt

b. What is the probability that A would pick four clubs? 1 pt

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d. How many possible draws would own exactly two kings and two clubs? 1 pt

Exercise 5 : Conditional probabilities (3 points)

A territorial organization conducted a study in the arts and crafts (fr: artisanat) sector. Its intention is to allocate a subsidy (fr: subvention) to craftspeople (fr: artisans) who request it, and preferably to those who have a professional development project and who wish to carry it out.

* 30 % of craftspeople really have such a project and, among them, 92% ask for a subsidy (and will get it);

* Among the other 70%, one fifth ask for a subsidy, will get it, but won't carry some project out.

If at least two thirds of the allocated subsidies will contribute to real and accomplished projects, the organization will consider its "subsidy operation" as a success.

1) According to the results of the study, build, as you wish: a probabilistic choice tree or a contingency table

(you may then choose a total number of 1000 craftspeople). 1.5 pt

2) Calculate the probability that a craftsperson would carry his (or her) project out, given that the

organization allocated a subsidy to him (or her). Can the "subsidy operation" be regarded as a success?

1,5 pt

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Exercise 6 : Random variable (4 points)

A company is used to manufacture three products A, B and C, and is forecasting its commercialisation costs.

A represents 20 % of the production, B represents 45%, and C represents 35 %. Typical commercialisation costs, for each produced unit of A, B, or C, are in the same order: €30, €36, €42, except in 30 % cases (same for A, B, or C), where this cost rises by €6 (extra costs for exports).

1) Build a probabilistic choice tree that displays the six different possible situations of costs. 0.5 pt

2) Give the probability distribution of the variable X : "commercialisation cost of a unit". 1 pt

3) What is the probability that X may exceed €40? 0.5 pt

4) Calculate the expectation of X. Interpret its value. 1 pt

5) Give an estimate of the commercialisation cost of 5,000 produced units, following the information given

by the directions of this exercise. 1 pt

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