IUT of Saint-Etienne – Sales and Marketing department Mr Ferraris Prom 2018-2020 10/2019 MATHEMATICS – 3

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IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2018-2020 10/2019

MATHEMATICS – 3

^rd

semester, Test 1 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: (4.5 points)

In a bag are placed 5 white balls and 10 black ones. You have to draw three of the fifteen, simultaneously and at random. If you get three white balls, you win € 100; if you get only two white ones, you win € 10; only one white makes you win or lose nothing; with three black balls, you lose € 20.

1) Let Y be the random variable of the number of white balls that you will get among the three drawn balls.

Explain the type of this variable’s probability distribution, then give its parameters. 1 pt

2) We name X the random variable « gain in € after one game ».

a. By building a table, give the probability distribution of X (show the detailed calculation of at least one

of the probabilities). 1 pt

b. Give the expected value and the standard deviation of X. Comment them, quickly. 1 pt

Full Name : Group : B2

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c. Considering 10,000 games to be played, what global gain is more likely than any other one? Also, give a confidence interval of the global gain, thanks to the knowledge of X’s standard deviation 1.5 pt

Exercise 2: (4 points)

A telecommunications agency performs a test on the connection status of 20 lines. A line has a 1 in 10 chance of being disconnected, and therefore 9 in 10 chances of not having a problem. The random variable X gives the number of problems encountered, out of the 20 lines tested.

1) Give, and justify, the probability distribution of X. 1 pt

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2) a. What is the most likely number of problems after 20 tests? Give its probability. 0.5 pt

b. What is the probability that less than two problems would be encountered? 0.5 pt

3) 8 groups of tests, 20 lines each, have to be performed. In this question, we name « success » the fact that one group of tests leads to less than two problems. The random variable Y is the number of successes among the 8 groups.

a. What is the most likely value of Y? 0.5 pt

b. What is the probability that at least one success would be obtained? 0.5 pt

c. Instead of 8, how many groups would have to be tested so that the probability of having at least one

success exceeds 99.9%? 1 pt

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Exercise 3: (3.5 points)

1) The random variable U is distributed by the standard normal law. Determine:

a. p(U < 2.28) ; b. p(U > 1.44) ; c. p(–1.5 < U < 1.5) 1 pt

d. the value u0 such that p(U > u0) = 0.8 1 pt

2) The random variable X is distributed by the normal law with a mean of 126 and a standard deviation that values 6. Determine:

a. p(X > 135) 0.5 pt

b. the value x0 such that p(X > x0) = 0.01 1 pt

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An internal study conducted in a large company revealed that 75% of employees are willing to work overtime.

A new agency is to be opened, in which 80 employees of the company will have to work. Of these 80 employees, we are interested in the number of those who will agree to work overtime, and we note X the random variable associated with this number.

1) a. Show that X is distributed by a binomial law and give its parameters. 0.5 pt

b. Show that this law can be approximated by a normal law and give its parameters. 0.5 pt

2) a. Calculate with this normal law the probability that less than 55 people will accept overtime. 1 pt

c. Similarly, calculate the probability that this number of people will be between 55 and 65. 1 pt

3) What is the minimum number of employees who would be likely to accept overtime, which can be given

with a 99% confidence level? 1,5 pt

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A device in a production line fills vials (fr : flacons), which are sold as containing 100 mg of product. But the machine is not perfect: the real quantity X introduced into a vial is a random variable of normal distribution N(m , σ = 1.1 mg), m being adjustable by the operator.

1) The machine is set to m = 101.2 mg. Out of 1000 vials, how many will actually contain less than 100 mg of

product? 1.5 pt

2) On what mean should the machine be set so that less than 1% of the vials contain less than 100 mg of

product? 2 pts

____________________ TEST END ____________________

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IUT TC Form of the test 1, 3rd semester MATHEMATICS

Probability distributions

Hypergeometric

H

(n, a, N) n: number of draws; a: number of "success" individuals; N: population's size;

k: number of desired successes, after n draws

approx. Hypergeometric by

binomial: if N ≥ 20n. We set p = a/N Binomial

B

(n, p) n: number of draws; p, q: probabilities of success, of failure

approx. binomial by Poisson:

if n ≥ 30 and p < 0.1 and np < 10. We set λ^{= np} Poisson

P

₍_λ₎

Approximation of

B

(n, p) by

N

₍_µ_,_σ) : if n ≥ 30, np ≥ 5, nq ≥ 5 ; we set µ^{= np and}σ⁼ ^npq Approximation of

P

₍_λ_{) by}

N

₍_µ_,_σ_{) : if}_λ_≥ 20 ; we set µ⁼λ^andσ⁼

λ

scheme:

H

(n, a, N)

B

_{(n, p)}

if N > 20n

if n ≥ 30 if n ≥ 30 if np ≥ 5 if p < 0.1 if nq ≥ 5

if np < 10 with µ^{= np and}σ⁼ ^npq

P

₍_λ)

N

₍_µ_,_σ₎

with λ = np

if λ≥ 20 with µ⁼λ^andσ⁼

λ

( )

p X =k =C^k_np q^k ^{n k}⁻

( ) ( )

2

N N

V N N 1

a a n

X =n − −

−

( )

_!

p e

k

X k

k

= = −^λ λ ^E

( )

^X ⁼

^λ

^; ^V

( )

^X ⁼

^λ (

X k

)

^k^a n^{n k}^a

−

× −

= = ^N

N

C C

p C ^E

( )

N X =na

( )

E X =np ^V

( )

^X ⁼^npq

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Tables

Poisson's law table Table of probabilities: values p(X = k) for several Poisson's distributions

λ ^0.1 ^0.2 ^0.3 ^0.4 ^0.5 ^0.6 ^0.7 ^0.8 ^0.9 k 0 0.90484 0.81873 0.74082 0.67032 0.60653 0.54881 0.49659 0.44933 0.40657

1 0.09048 0.16375 0.22225 0.26813 0.30327 0.32929 0.34761 0.35946 0.36591 2 0.00452 0.01637 0.03334 0.05363 0.07582 0.09879 0.12166 0.14379 0.16466 3 0.00015 0.00109 0.00333 0.00715 0.01264 0.01976 0.02839 0.03834 0.04940 4 0.00000 0.00005 0.00025 0.00072 0.00158 0.00296 0.00497 0.00767 0.01111 5 0.00000 0.00000 0.00002 0.00006 0.00016 0.00036 0.00070 0.00123 0.00200 6 0.00000 0.00000 0.00000 0.00000 0.00001 0.00004 0.00008 0.00016 0.00030

λ ¹ ^1.5 ² ^2.5 ³ ^3.5 ⁴ ^4.5 ⁵

k 0 0.36788 0.22313 0.13534 0.08208 0.04979 0.03020 0.01832 0.01111 0.00674 1 0.36788 0.33470 0.27067 0.20521 0.14936 0.10569 0.07326 0.04999 0.03369 2 0.18394 0.25102 0.27067 0.25652 0.22404 0.18496 0.14653 0.11248 0.08422 3 0.06131 0.12551 0.18045 0.21376 0.22404 0.21579 0.19537 0.16872 0.14037 4 0.01533 0.04707 0.09022 0.13360 0.16803 0.18881 0.19537 0.18981 0.17547 5 0.00307 0.01412 0.03609 0.06680 0.10082 0.13217 0.15629 0.17083 0.17547 6 0.00051 0.00353 0.01203 0.02783 0.05041 0.07710 0.10420 0.12812 0.14622 7 0.00007 0.00076 0.00344 0.00994 0.02160 0.03855 0.05954 0.08236 0.10444 8 0.00001 0.00014 0.00086 0.00311 0.00810 0.01687 0.02977 0.04633 0.06528 9 0.00000 0.00002 0.00019 0.00086 0.00270 0.00656 0.01323 0.02316 0.03627 10 0.00000 0.00000 0.00004 0.00022 0.00081 0.00230 0.00529 0.01042 0.01813 11 0.00000 0.00000 0.00001 0.00005 0.00022 0.00073 0.00192 0.00426 0.00824 12 0.00000 0.00000 0.00000 0.00001 0.00006 0.00021 0.00064 0.00160 0.00343

λ ^5.5 ⁶ ^6.5 ⁷ ^7.5 ⁸ ^8.5 ⁹ ^9.5 ¹⁰

k 0 0.00409 0.00248 0.00150 0.00091 0.00055 0.00034 0.00020 0.00012 0.00007 0.00005 1 0.02248 0.01487 0.00977 0.00638 0.00415 0.00268 0.00173 0.00111 0.00071 0.00045 2 0.06181 0.04462 0.03176 0.02234 0.01556 0.01073 0.00735 0.00500 0.00338 0.00227 3 0.11332 0.08924 0.06881 0.05213 0.03889 0.02863 0.02083 0.01499 0.01070 0.00757 4 0.15582 0.13385 0.11182 0.09123 0.07292 0.05725 0.04425 0.03374 0.02540 0.01892 5 0.17140 0.16062 0.14537 0.12772 0.10937 0.09160 0.07523 0.06073 0.04827 0.03783 6 0.15712 0.16062 0.15748 0.14900 0.13672 0.12214 0.10658 0.09109 0.07642 0.06306 7 0.12345 0.13768 0.14623 0.14900 0.14648 0.13959 0.12942 0.11712 0.10371 0.09008 8 0.08487 0.10326 0.11882 0.13038 0.13733 0.13959 0.13751 0.13176 0.12316 0.11260 9 0.05187 0.06884 0.08581 0.10140 0.11444 0.12408 0.12987 0.13176 0.13000 0.12511 10 ^0.02853 ^0.04130 0.05578 0.07098 0.08583 0.09926 0.11039 0.11858 0.12350 0.12511 11 ^0.01426 ^0.02253 0.03296 0.04517 0.05852 0.07219 0.08530 0.09702 0.10666 0.11374 12 0.00654 0.01126 0.01785 0.02635 0.03658 0.04813 0.06042 0.07277 0.08444 0.09478 13 0.00277 0.00520 0.00893 0.01419 0.02110 0.02962 0.03951 0.05038 0.06171 0.07291 14 0.00109 0.00223 0.00414 0.00709 0.01130 0.01692 0.02399 0.03238 0.04187 0.05208 15 0.00040 0.00089 0.00180 0.00331 0.00565 0.00903 0.01359 0.01943 0.02652 0.03472 16 0.00014 0.00033 0.00073 0.00145 0.00265 0.00451 0.00722 0.01093 0.01575 0.02170

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Standard normal law table gives probabilities p(U < u) How to get u from x : x

u µ

σ⁻

=

u 0 1 2 3 4 5 6 7 8 9

0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900 3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929 3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950 3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965 3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976 3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983 3.6 0.999841 0.999847 0.999853 0.999858 0.999864 0.999869 0.999874 0.999879 0.999883 0.999888 3.7 0.999892 0.999896 0.999900 0.999904 0.999908 0.999912 0.999915 0.999918 0.999922 0.999925 3.8 0.999928 0.999931 0.999933 0.999936 0.999938 0.999941 0.999943 0.999946 0.999948 0.999950 3.9 0.9999519 0.9999539 0.9999557 0.9999575 0.9999593 0.9999609 0.9999625 0.9999641 0.9999655 0.9999670

U u

p(U < u)