IUT of Saint-Etienne – Sales and Marketing department Mr. Ferraris Prom 2016-2018 26/10/2017 MATHEMATICS – 3

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IUT of Saint-Etienne – Sales and Marketing department

Mr. Ferraris Prom 2016-2018 26/10/2017

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)

A bag contains 16 red coins and 4 green ones. A game consists in a simultaneous draw of ten coins, at random.

The player wins in case he gets at least three green coins.

1) After having determined the probability distribution of X, number of green coins among the ten, calculate

the probability to win. 2 pts

2) To be allowed to play once, you have to bet €2. The game owner settled the following gains:

€20 for 4 green coins, €4 for 3 green coins, and no gain in case of any other outcome. How much money

can the owner expect to win until 10,000 played games? 2 pts

Full Name : Group : B2

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Exercise 2 (8.5 points)

In a supermarket, it has been stated that the pass-through of one customer in twenty causes a problem at the checkout (caisse) and blocks the line (file d'attente) for a while. At any given time, for a given checkout, the probability of occurrence of a problem is then 0.05. After having taken your products, you go choose a checkout and enter a line that consists of n customers in front of you. The variable X describes the number of problems that may occur and waste your time due to the pass-through of these n clients.

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

2) In this question, let's set n = 5.

a. What is the probability that at least one problem would occur before your turn? 1 pt

b. Given that the supermarket opens 10 checkouts, and that there is an average of 5 clients per checkout, give an estimate of the number of checkouts blocked by a problem, at any time. 1 pt

3) a. What is the maximum number of clients that have to be in front of you in the line, so that there are more than 90% chances that no problem would occur before your turn? 1 pt

b. What is the minimum number of clients that have to be in front of you in the line, so that there are more than 50% chances that at least one problem would occur before your turn? 1.5 pt

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4) We assume that, on average, one checkout registers 100 payments per day. In this question, we aim to use a Poisson distribution, and we will name Y the variable daily number of problems suffered by a checkout.

a. Show that Y can be distributed by a Poisson's law, whose parameter will be given. 1 pt

b. What is the probability that, in a day, less than 3 problems occur at a checkout? 1 pt

c. What is the minimum number of problems that have less than 1% chance to be exceeded? 1 pt

Exercise 3 (7.5 points)

A raspberry's (framboise) mass forms a variable X, well-modeled by the law

N

(4 grams ; 0.5 g).

1) Determine p(X < 3) ; p(3 < X < 5). 1 pt

2) The raspberries whose mass is less than 3 grams might be removed by the producer, before the delivery. In a 40 kg batch, representing around 10,000 raspberries, how many of them would be removed? 1 pt

3) On a production line, it's planned to sort raspberries by mass: the lightest 20% will be packaged separately, sold at a reduced price and renamed "raspbabies". What is the maximum mass of a raspbaby? 2 pts

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4) From a production of 10,000, a sample of 300 raspberries has to be drawn. Y is the possible number of raspbabies among the 300.

a. Explain why a binomial distribution would correctly describe Y, and give its parameters. 1 pt

b. Explain why a normal distribution could correctly replace this binomial one, and then give the

parameters of this new normal one. 1 pt

c. Using both distributions, give the probability of getting between 55 and 65 (included) raspbabies, and then the probability of getting exactly 60 raspbabies. Comment the visible differences between the

results coming from different laws. 1.5 pt

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

Probability distributions

Hypergeométric

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)