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Ïîäáð.2 èãðàëüíûõ êîñòè. Íàéòè âåð-òü ñîá-é:                 2.3 áåëûõ øàðà è 2 ÷åðíûõ. Íàéòè âåð-òü        3. Èç êîëîäû êàðò â 36 øò. äîñòàåòñÿ 4 êàðòû.      4. Èç ÿùèêà, ñîäåðæàùåãî 5 áåë,2 ÷åðí,3 êðàñí,

A-âûïàäåò 2 äâîéêè                                                                    ÷òî îáà øàðà áåëûå.                                            Íàéòè âåð-òü, ÷òî õîòÿ áû îäíà èç íèõ ÒÓÇ.           èçâëåêàþò 2 øàðèêà. Íàéòè âåð-òü:À-áåë è

B-âûïàäåò õîòÿ áû îäíà 1                                                       Ðåøåíèå: P(A)=na/n, na-÷èñëî èñõîäîâ,          Ðåøåíèå: P(A)=1-P(A)                                                êðàñ. B- íå ìåíåå îäíîãî êðàñ.

C-âûïàäóò 2 íå÷åòíûå öèôðû                                                   êîãäà À ïðîèçîøëî, n - îáù.÷èñëî ñîá.              P(A)=1-(C{4 32}/C{4 36})=0.4                               Ðåøåíèå:  P(A)=na/n

D- ïðè ïåðâîì ïîäáðàñ.2, ïðè âòîðîì – 5                                n=C{2 5} na=C{2 3}                                                                                                                                  n-C{2 10}    na=C{2 8}

Ðåøåíèå: n=6*6=36-îáù. ÷èñëî èñõîäîâ                                                                                                                                                                                                    P(A)=  (C{1 5}*C{1 3})/C{2 10}=8/45                                                                                                                                                        

P(A)=(1*1)/36=1/36               

P(B)åñëè çàäàíî íàéòè âåð-òü õîòÿ áû îäíó,

òî íàäî ïåðåéòè ê ïðîòèâîïîëîæíîìó ñîáûòèþ

A –ïðîòèâîïîë. Ñîá., ò.å. íå îäíîé

P(B)=1-P(B)

P(B)=(5*5)/36=25/36

P(B)=1-25/36=11/36

P(C)=(3*3)/36=1/4

P(D)=1/36

5. Ëîòåðåÿ 5 èç 36.Íàéòè âåð-òü óãàäàòü 4 öèôðû. Ðåøåíèå:P(A)=(C{4 5}*C {1  31})/C{5 36}

 

6. Â ëèôò 9 ýòàæíîãî äîìà çàøëî òðîå. Íàéòè âåð-òü, ÷òî îíè âûéäóò íà ðàçí.ýòàæàõ Ðåøåíèå: P(A)=(8*7*6)/(8*8*8)=56/64 

7.Ïîäáðàñ.2 èãðàëüí. Êîñòè. Îïðåä. ßâëÿåòñÿ ëè  ñîá.À çàâèñèìûì îò ñîá.Â, åñëè À-ïðè ïåðâîì ïîäáðàñûâàíèè âûïàëî 3 î÷êà, Â-ñóììà î÷êîâ ïðè äâóõ ïîäáðàñûâàíèÿõ=8. Ðåøåíèå:P(A)=1/6, P(B)=5/36,  P(A*B)=1/36  Âûâîä: 1/36 íå ðàâíà 1/6*5/36 ñîáûòèÿ À è  çàâèñèìû.

8.Íàéòè âåð-òü óãàäàòü ñ äâóõ ïîïûòîê 4-õ çíà÷íûé êîä èç öèôð. Ðåøåíèå: P(A)=1/10â 4 ñòåïåíè. A=A1+A1*A2-åñëè íå óãàäàëè ñ ïåðâîé ïîïûòêè, à ñî âòîðîé óãàäàëè. P(A)=P(A1+A1*A2)=P(A1)+P (A1*A2 )-P(A1+A1*A2). P(A1)=1/10â4 , P(A1\A2)=1/(10â4-1), P(A1*A2)=P(A1)*P(A2\A1)=(1-(1/10â4))*(1/10â4-1). P(A)=1/10â4+(1-(1/10â4))*(1/10â4-1)=2/10â4

9.Èìååòñÿ 7 òåííèñíûõ ìÿ÷åé 4-íîâ. 3 –ñòàð.Êàêîâà âåð-òü, ÷òî íå áîëåå 2 èç íèõ ñòàðûå, åñëè äîñòàåòñÿ ïî 4 ìÿ÷à.Ðåøåíèå:P(A)=P("0")+P("1")+P("2")=1-P("3")-P("4")=1-P4(3); n=4; p=3/7; q=4/7; 1-C{3 4}*(3/7)â3ñòåïåíè*(4/7)â1ñòåïåíèæ Ñ{m n}=4; 1-4*(27*4/49*49)=0.82

10. Èç ïàðòèè, ñîäåðæàùåé 20 èçäåëèé, ñëó÷àéíûì îáðàçîì âûáèðàþò 5. Êàêîâà âåð-òü, ÷òî íå ìåíåå 2-õ èçäåëèé èç âûáðàííûõ îêàæóòñÿ áðàêîâàííûìè, åñëè áðàê ñîñò.25%.Ðåøåíèå: n=5; p=1/4; q=3/4

P(A)=1-P("0")-P("1"); P(A)=1-C{0 5}*1/4*3/4â5ñòåïåíè – Ñ{1 5}*1/4*3/4â4ñòåïåíè

11. ×òî áîëåå âåðîÿòíî: âûèãðàòü ó ðàâíîñèëüíîãî ïðîòèâíèêà 2 ïàðòèè èç 3 èëè 3 ïàðòèè èç 5.Ðåøåíèå: ½, ò.ê. ïðîòèâíèê ðàâíîñèëüíûé. P(A)=C{2 3}*1/2â3ñòåïåíè; P(B)=C{3 5}*1/2â5ñòåïåíè=5/16; 3/8<5/16.

12. Äâîå ïîî÷åðåäíî ïîäáðàñûâàþò ìîíåòó, âûéãðàåò òîò, ó êîãî âûïàäåò ãåðá ðàíüøå. Íàéòè âåð-òü âûéãðûøà ä.êàæä.èç èãðîêîâ.Ðåøåíèå: P(1)=1/2+1/2*1/2*1/2+1/2â4ñòåïåíè*1/2…Ãåîìåòðè÷åñêàÿ ïðîãðåññèÿ a1=1/2; P(A)=2/3; q=1/4; P(2)=1/2*1/2+1/2â4ñòåïåíè+1/2â6ñòåïåíè; a0=1/4; q=1/4; P(B)=1/3

13. Äàí ðÿä ðàñïðåäåëåíèÿ                                                                        P(X=-1)=1/2/ Íàéòè M[x] b D[X]. Ðåøåíèå:  M[x]=-1*1/2+0*1/4+2*1/4=0; D[X]=-sqrt0+1*1/2+0*1/4+4*1/4=1.5>0 ñë-íî âåðíî.

14. Äàíà äèñêðåòíàÿ ÑÂ, êîò.ïðèíèìàåò çíà÷åíèÿ: -2, -1, 0, 2  ñ âåð-òüþ:1/3, 1/6, ¼ , p. Âû÷èñëèòü M[2x-x*x]. Ðåøåíèå: M[cx+b]=c*M[x]+b; M[x+y]=M[x]+M[y]; M[2x-x*x]=2M[x]-M[x*x]; 1/3+1/6+1/4+p=1 îòñþäà p=1/4. M[x]=-2*1/3+-1*1/6+2*1/4=1/3; M[x*x]=ñóììàx*x*p=4*1/3+1/6+1=15/6; M[2x-x*x]=2M[x]-M[x*x=-19/6

15. Äàíà ÑÂ ñ ðàâíîìåðíûì äèñêðåòíûì ðàñïðåäåëåíèåì.                                                                                                 M[x]=1/2. Íàéòè M[(1-x)â êâàäðàòå]. Ðåøåíèå: ñóììà 1/n*4=1; 1/n=1/4; M[x]=-1*1/4+a*1/4+2*1/4=1/2 ; ¼(-1+a+2)=1/2 ; a=1; M[(1-x)â êâàäðàòå]=M[(1-2x+x*x)]=1-2*1/2+M[x*x]=1-1+1.5=1.5; M[x]=1/2; M[x*x]=1*1/4+1*1/4+4*1/4=1.5

16. Äàíî 2 ÑÂ. P[X>=Y]=?                                                                                                        Ðåøåíèå: P(X>+Y)=P(X-Y>=0);                                                                           ò.ê. –1, òî èõ ìîæíî îáúåäèíèòü. Êàê ô-öèÿ, ïîäñòàâëÿåì çíà÷åíèÿ X & Y. P(x-y=-1)= P(x=1;y=0), à ò.ê. ýòè äâà ñîá ïðîèñõ.îäíîâðåìåííî,òî èñï.óìíîæåíèå 1/3*1/2=1/6; 1=1/6+1/6+1/3; åñëè  P(x-y>+0)   P(x-y-1)=1/3

17. Äàíà ÑÂ ñ áèíîì.ðàñïð. è ïàð.(1/2;3)Íàéòè M[x*x-3x] & D[3x-4]=? Ðåøåíèå: M[X*x-3x]=M[x*x]-3M[x]=n*p(1-p)=3*1/2(1-1/2)=3/4; D(x)=M[x*x]-(M[x]) â êâàäðàòå; M[x*x]=D(x)+ (M[x])â êâàäðàòå; M[x*x]=3/4+3/2â êâàäðàòå=3; M[x]=3/2; M[x*x-3x]=3-9/2=3/2; D[3x-4]=9D[x]ïîñâ-âó =3/4*9=27/4

Ïîäáð.2 èãðàëüíûõ êîñòè. Íàéòè âåð-òü ñîá-é:                 2.3 áåëûõ øàðà è 2 ÷åðíûõ. Íàéòè âåð-òü        3. Èç êîëîäû êàðò â 36 øò. äîñòàåòñÿ 4 êàðòû.      4. Èç ÿùèêà, ñîäåðæàùåãî 5 áåë,2 ÷åðí,3 êðàñí, A-âûïàäåò 2 äâîéêè                   

 

 

 

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