IUT of Saint-Etienne – Sales and Marketing department Mr Ferraris Prom 2018-2020 05/2019 MATHEMATICS – 2

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IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2018-2020 05/2019

MATHEMATICS – 2

^nd

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

The graphic calculator is allowed. Any personal sheet is forbidden.

Your work has to be written down inside this document.

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

Your rounded results will show at least four significant figures.

Exercise 1 - Sets (3 points)

Here, A and B are two subsets of a set E.

1) To what does the set A refer? 0.5 pt

2) What does the sentence "A and B are mutually exclusive" mean? 0.5 pt

3) What does the sentence "A and B form a partition of E" mean? 1 pt

4) Simplify, on giving details, the expression



^A^^B



^



^A^^B



^. ^{1 pt}

Full Name : Group : B1

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Exercise 2 - Cardinal numbers (3 points)

A store offers two relatively complementary objects A and B. To study the purchasing behaviour on these objects, the management recruited a trainee, who gave him/her the following information: on a number of the last receipts, it appeared that 127 people bought both objects, 64 bought only one of them (those who bought only the object A represent a quarter of them) and 109 bought neither A nor B

1) These indications are incomplete. Organize a contingency table crossing the purchase (and non-purchase) quantities of the objects A and B, then complete this table. 1.5 pt

2) Calculate:

a. the rate of the clients who bought the object A. 0.5 pt

b. the rate of the clients who only bought the object A. 0,5 pt

c. the rate of the clients who bought the object B, among those who bought the object A. 0.5 pt

Exercise 3 - Combinatorics (4.5 points) – the three questions are independent

1) A bag contains eight coins numbered from 1 to 8. We have to draw three of them, successively and without putting back, so as to create a number (like 327 for instance). How many numbers are possible?

1 pt

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2) While walking the streets of a big city, one has the choice at each crossroads to turn to the left, to turn to the right, or to go straight. Our route must go through 6 intersections. How many different routes could

we do? 1 pt

3) 80 candidates are competing in a first round of a game. Only 5 of them will be qualified for the second round.

a. How many possible qualified groups are there? 1 pt

b. If the candidates consist of 30 men and 50 women, how many groups of five people can be composed

with two men and three women? 1.5 pt

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Exercise 4 - Probabilities (4 points)

We are in the middle of winter. The chances that it will snow tomorrow (event N) are estimated at 2 out of 3. If it snows tomorrow, 120 students out of 160 of the promotion will be present in class. If it does not snow, 150 students will be present. The next day, in the list of 160 students, a name will be drawn at random. Let’s name A the event "the student is absent".

1) Built a probabilistic choice tree with the events A and N. 1 pt

2) What is the probability that a name drawn at random corresponds to a present student? 1 pt

3) Given that a student is present, what is the probability that it’s snowing? 1.5 pt

4) Are the events N and A independent? 1 pt

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Exercise 5 - random variable and probability distribution (5 points)

A new scratch game consists in a ticket on which are located 30 golden discs. Four of these discs are "winners":

if we scratched them, we would discover a star; the other 26 are "losers": no picture under the gilding (fr.:

“dorure”). To play, we must scratch three discs at random, not more, not less.

1) Four events are possible in the end: to get three stars, or two, or one, or none. Explain why their probabilities are in the same order: 0.0009852; 0.03842; 0.3202; 0.6404. 1.5 pt

2) You have to spend €2 to buy a ticket. The fixed gains are the following: €500 with 3 stars, €10 with 2 stars,

€2 with one star and €0 if no star is discovered. We set X as the variable “gain after a game, minus the

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b. Give its expected value E(X) and its standard deviation (X). Explain these values in details. 2 pts

c. On buying 500 tickets and scratching all of them, what average global gain can be expected? 0.5 pt

d. What percentage of the players’ expenses is actually redistributed in gains? 0.5 pt

____________________ TEST END ____________________

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IUT TC MATHEMATICS FORM for COMBINATORICS and PROBABILITIES

3) Combinatorics