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Category: Probability and Statistics

On-dispose-de-N-1-urnes-l-urne-U-k-contient-k-boules-blanches-et-N-k-boules-noires-on-tire-successivement-sans-remise-n-boules-de-l-urne-et-on-note-An-l-evenement-choisir-n-boules-noires-lors-des

Question Number 145134 by ArielVyny last updated on 02/Jul/21 $${On}\:{dispose}\:{de}\:{N}+\mathrm{1}\:{urnes}.{l}'{urne}\:{U}_{{k}} \\ $$$${contient}\:{k}\:{boules}\:{blanches}\:{et}\:{N}−{k}\:{boules} \\ $$$${noires}.{on}\:{tire}\:{successivement}\:{sans}\: \\ $$$${remise}\:{n}\:{boules}\:{de}\:{l}'{urne}\:{et}\:{on}\:{note}\: \\ $$$${An}\:{l}'{evenement}\:''{choisir}\:{n}\:{boules}\:{noires} \\ $$$${lors}\:{des}\:{n}\:{premiers}\:{tirages}''.\:{Determiner} \\ $$$${P}\left({An}\right).\:{on}\:{notera}\:{U}_{{k}} =''{choisir}\:{l}'{urne}\:{k}'' \\ $$…

Soit-X-une-variable-aleatoire-de-loi-geometrique-de-parametre-p-0-1-calculer-P-X-4-

Question Number 145136 by ArielVyny last updated on 02/Jul/21 $${Soit}\:{X}\:{une}\:{variable}\:{aleatoire}\:{de}\:{loi} \\ $$$$\left.{geometrique}\:{de}\:{parametre}\:{p}\in\right]\mathrm{0}.\mathrm{1}\left[\right. \\ $$$${calculer}\:{P}\left(\left\{{X}\geqslant\mathrm{4}\right\}\right) \\ $$ Answered by Olaf_Thorendsen last updated on 03/Jul/21 $${P}\left({X}=\:{k}\right)\:=\:{q}^{{k}−\mathrm{1}} {p}\:=\:\left(\mathrm{1}−{p}\right)^{{k}−\mathrm{1}}…

A-debating-team-is-to-be-selected-from-a-group-of-8-boys-and-6-girls-in-how-many-ways-can-this-be-done-if-the-team-a-Must-be-either-all-boys-or-all-girls-b-Must-consist-of-two-boys-and-two-gir

Question Number 13336 by tawa tawa last updated on 19/May/17 $$\mathrm{A}\:\mathrm{debating}\:\mathrm{team}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{selected}\:\mathrm{from}\:\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{8}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{6}\:\mathrm{girls}.\:, \\ $$$$\mathrm{in}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{this}\:\mathrm{be}\:\mathrm{done}\:\mathrm{if}\:\mathrm{the}\:\mathrm{team}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Must}\:\mathrm{be}\:\mathrm{either}\:\mathrm{all}\:\mathrm{boys}\:\mathrm{or}\:\mathrm{all}\:\mathrm{girls} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Must}\:\mathrm{consist}\:\mathrm{of}\:\mathrm{two}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{two}\:\mathrm{girls} \\ $$ Commented by tawa tawa last updated…

A-card-is-drawn-at-random-from-a-well-shuffled-deck-of-52-cards-Find-the-probability-of-its-being-a-spade-or-a-king-

Question Number 21 by user1 last updated on 25/Jan/15 $$\mathrm{A}\:\mathrm{card}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{a}\:\mathrm{well} \\ $$$$\mathrm{shuffled}\:\mathrm{deck}\:\mathrm{of}\:\mathrm{52}\:\mathrm{cards}.\:\mathrm{Find}\:\mathrm{the}\:\: \\ $$$$\mathrm{probability}\:\mathrm{of}\:\mathrm{its}\:\mathrm{being}\:\mathrm{a}\:\mathrm{spade}\:\mathrm{or}\:\mathrm{a}\:\mathrm{king}. \\ $$ Answered by user1 last updated on 30/Oct/14 $$\mathrm{Let}\:{S}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sample}\:\mathrm{space}.\:{n}\left({S}\right)=\mathrm{52}.\: \\…

soit-X-i-i-1-2-n-une-suite-de-variable-aleartoire-independante-iid-suivant-la-loi-binomiale-B-n-p-montrer-que-X-n-converge-en-loi-vers-E-X-2-montrer-que-1-n-i-1-2-X-i-2

Question Number 143593 by pticantor last updated on 16/Jun/21 $${soit}\:\left(\boldsymbol{{X}}_{{i}} \right),{i}\in\left\{\mathrm{1},\mathrm{2},……,{n}\right\}\:{une}\:{suite}\:{de}\:{variable}\:{a}\boldsymbol{{leartoire}}\:\boldsymbol{{independante}} \\ $$$$\left(\boldsymbol{{iid}}\right)\:\boldsymbol{{suivant}}\:\boldsymbol{{la}}\:\boldsymbol{{loi}}\:\boldsymbol{{binomiale}}\:\boldsymbol{{B}}\left(\boldsymbol{{n}},\boldsymbol{{p}}\right)\: \\ $$$$\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\overset{\_} {\boldsymbol{{X}}}_{\boldsymbol{{n}}} \boldsymbol{{converge}}\:\boldsymbol{{en}}\:\boldsymbol{{loi}}\:\boldsymbol{{vers}}\:\boldsymbol{{E}}\left(\boldsymbol{{X}}\right) \\ $$$$\mathrm{2}−\:\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}}\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}} {\sum}}\boldsymbol{{X}}_{\boldsymbol{{i}}\:\:} ^{\mathrm{2}} \boldsymbol{{converge}}\:\boldsymbol{{en}}\:\boldsymbol{{loi}}\:\:\boldsymbol{{vers}}\:\:\:\:\:\:\:\:\boldsymbol{{E}}\:\left(\boldsymbol{{X}}^{\mathrm{2}} \right) \\…