IUT of Saint-Etienne – Sales and Marketing department
Mr Ferraris Prom 2020-2022 05/05/2021
MATHEMATICS – 2
ndsemester, 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 : combinatorics (7 points)
The three questions of this exercise are independent
1) Ten coins are to be tossed, each of which can give the result "heads" or "tails". How many possible outcomes
are there to this experiment? 1.5 pt
2) a. In how many ways can four students be chosen at random from a group of 25 students? 1.5 pt
b. If this group consists of 10 men and 15 women, what is the probability of having chosen only one woman,
randomly selecting four students? 1.5 pt
Full Name : Group : B1
3) We wish to determine the number of integers, between 0 and 9999, whose digits are different and which do not contain the digit 5. To do this, we will split up the problem according to the number of digits present:
we must count separately the numbers between 1000 and 9999, those between 100 and 999, those between 10 and 99, and the integers from 0 to 9. How many such numbers are there in all? 2.5 pts
Exercise 2 : probabilities (7 points)
A website offers a quizz with two series of questions one after the other. 60% of people answer the questions in the first series honourably (majority of correct answers). If this is the case, then the site chooses a second, more difficult series, in which only 30% of people have a majority of correct answers. On the contrary, if a person did not have an honourable result in the first round, then the site chooses a second, less difficult round, in which 50% of people have a majority of correct answers.
Event A : answer honourably to the first set of questions Event B : answer honourably to the second set of questions
1) Form either a tree or a table to represent the situation. 1.5 pt
2) If you take the quiz, how likely is it that...
a. you answer honourably to both series? 1 pt
b. you answer honourably to the second series? 1.5 pt
c. you answer honourably to the first series, given that you didn’t answer honourably to the second one?
1.5 pt
3) Points are accumulated in case of success: passing series A awards 5 points and passing series B awards 10 points. The random variable X is the total score that can be reached for players who take part in both series in succession.
a. What are the possible values of X ? 0.5 pt
b. Give the probability distribution of this random variable. 1 pt
Exercise 3 : probability distributions (6 points)
Let's play spin a wheel. It is divided into 50 sectors of the same size. If you hit sector n°1, you win €80; if you hit one of the sectors n°2 to n°10, you win €3; otherwise, you win nothing. To be able to play (once), you have to pay 3 €. X is the random variable referring to the winnings (net: remove the €3 bet) at the end of a game.
1) What is the probability distribution of X? 2 pts
2) a. Give the expected value and the standard deviation of X ; comment on the expectation. 2 pts
b. How can the value of the standard deviation be used? (give the meaningof this parameter, and then the way it can be used to forecast the global or the average gain in a certain number of games, number to
be chosen) 2 pts
____________________ TEST END ____________________
IUT TC MATHEMATICS FORM for COMBINATORICS and PROBABILITIES
* Confidence intervals of the overall value of the average value
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3) Combinatorics
