IUT of Saint-Etienne – Sales and Marketing department Mr Ferraris Prom 2017-2019 08/11/2018 MATHEMATICS – 3

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IUT of Saint-Etienne – Sales and Marketing department

Mr Ferraris Prom 2017-2019 08/11/2018

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 points)

In a group of 20 students, 14 had at least ten points in mathematics ("T" students).

You have to choose 10 students at random from this group. The random variable X is the number of "T"

students among these 10.

1) What is your expectation, in terms of "T" students among the 10 chosen? 1 pt

2) Give, after having justified, the probability distribution of X. 1.5 pt

3) What is the probability to get 7 "T" students among the 10? 1.5 pt

Full Name : Group : B2

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Exercise 2 (5.5 points)

A company manufactures bolts (fr.: boulons). At the production line's exit, a sample is taken at random and verified, showing that 1.5 % of the production is defective.

1) Bolts are packed in boxes: 20 bolts in each box. Given that the whole produced quantity is very big compared to a 20-bolt sample, we are allowed to consider the box as made by a draw with putting back. Let be X the random variable "number of defective bolts in a box".

a. What is the probability distribution of X? Justify. 1 pt

b. A box can be sent back by a client, and will have to be reimbursed, if it contains at least two defective bolts. What is the probability this situation would happen? 2 pts

c. In a total number of 1000 sold boxes, how many could the seller expect to reimburse? 0.5 pt

2) According to these former results, let's think about the number of boxes that would have to be reimbursed, in a batch containing 100 boxes. This number is variable and will be denoted Y. Let's fix its expectation at 4: on average, 4 boxes have to be reimbursed every 100 sold ones.

a. Justify that Y can be considered as distributed by a Poisson's law, which parameter is then 4. 1 pt

b. The situation will be considered as problematic for the company as soon as at least 8 boxes have to be

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Exercise 3 (7.5 points)

A company that sells office items decides to send its catalogue to 600 societies. On average, considering all the French societies (much more than 20 times 600), 10% of them place an order after having received such a catalogue. The variable X describes the number of societies that will place an order, among the 600.

1) By creating and analysing the probability distribution of X, show that it can be described by this normal one:

N

(60 , 7.348) 2 pts

2) Thanks to this normal distribution, determine:

a. the probability that more than 70 societies place an order. 1.5 pt

b. the probability that at least 55 societies place an order. 1.5 pt

3) Instead of 600, how many catalogues would have to be sent, at least, so that the probability of the event in

question 2b would reach 95% ? 2.5 pts

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Exercise 4 (3 points)

Only using the form below (standard normal law table), determine with

N

_{(0 , 1):}

a. p(U < 1.64). 1 pt

b. p(U < – 0.77). 1 pt

c. p(-2 < U < 2). 1 pt

____________________ 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 and npq ≥ 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 p < 0,1 if npq ≥ 5 if np< 10

P

₍_λ)

N

₍_µ_,_σ₎

with λ = np with µ^{= np and}σ⁼ ^npq 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)