Òåîðèÿ âåðîÿòíîñòåé
1.
Íåçàâèñèìî äðóã îò äðóãà 10 ÷åë. Ñàäÿòñÿ â ïîåçä,
ñîäåðæàùèé 15 âàãîíîâ.
Âåðîÿòíîñòü òîãî, ÷òî âñå îíè ïîåäóò â ðàçíûõ âàãîíàõ?
Ð= ÷èñëî áëèçêèõ èõîäîâ = 15….14…….-
6 = 15 ! -2
×èñëî ýëåìåíò. èñõîäîâ 15*15*15…15
5 ! » 1,88 * 1å
10 ðàç 50
15 _____________________________________
2.  ýëåêòðè÷åñêîé öåïè ïîñëåäîâàòåëüíî
âêëþ÷åíû 3 ýëåìåíòà, ðàáîòàþùèå
íåçàâèñèìî äðóã îò äðóãà. Èõ âåð-òü îòêàçîâ
ðàâíû 1 49 1 .
Íàéòè âåðîÿòíîñòü òîãî, ÷òî òîêà íå
áóäåò? 50 ; 50 ; 4
-- - -
À –òîê åñòü
Ài – i-é ïðèáîð íå èñïðàâåí
Ð (À1) = 49 Ð (À2)= 1 Ð ( À3) = 3
50 ; 50
; 4
_
Ð (À)=1-Ð(À) = 1-Ð (À1 À2 À3 ) = 1-Ð (À1) Ð (À2)* Ð (À3) = 1- 49 * 1- 3
= 9,753
50 50 4 10,000
____________________________________________________________________________________________
3. Âåð-òü ïîïàäàíèÿ õîòÿ áû ðàç â ìèøåíü ïðè
12-òè âûñòðåëàõ ðàâíî 41 .
Íàéäèòå âåð-òü ïîïàäàíèÿ ïðè îäíîì
âûñòðåëå? 50
Ài – óñïåøíûé i – âûñòðåë
_________
Ð = 41 = 1-Ð ( À1 …..À12) – íå
ïîïàëè íè â îäíîì ñëó÷àå èç 12-è âûñòðåëîâ =
50
__
__ _ 12
12
= 1 – Ð (À1) …..Ð (À12) = 1 – Ð (À1) ; 41 = 1-Ð (À1)
50
Íàéòè Ð (À1)
_ 12
Ð (À1) = 1- 41 = 9
50 50
_ 12__
Ð (À1) = Ö 9
50
_ 12__
Ð (À1) = 1-Ð (À1) = 1 - Ö 9 » 0,133
50
___________________________________________
4.
Èìåþòñÿ 28 áèëåòîâ, íà êàæäîì èç êîòîðûõ íàïèñàíî
óñëîâèå íåñêîëüêèõ
çàäà÷.  13 áèëåòàõ çàäà÷è ïî ñòàòèñòèêå, à â
îñòàëüíûõ 15 – çàäà÷è ïî òåîðèè
âåðîÿòíîñòè. 3 ñòóäåíòà âûáèðàþò íà óäà÷ó ïî
îäíîìó áèëåòó. Íàéòè âåðîÿòíîñòü
òîãî, ÷òî õîòÿ áû îäíîìó èç ñòóäåíòîâ íå
äîñòàíåòñÿ çàäà÷à ïî òåîðèè âåðîÿòíîñòè.
Ài –ñòóäåíòó äîñòàíåòñÿ
çàäà÷à ïî òåîðèè âåðîÿòíîñòè
À – âñåì äîñòàíåòñÿ çàäà÷à ïî òåîð. âåðîÿò.
À = À1 À2 À3
À – õîòÿ áû îäíîìó íå äîñòàíåòñÿ çàäà÷à ïî
òåîð.âåðîÿò.
_
Ð (À) = 1 – Ð(À) = 1- Ð (À1 À2 À3) = 1 – Ð *(À3) * Ð (À1 À2) = 1-Ð *(À3) * Ð *
À1À2 À1À2 À1
*(À2)*Ð
(À1)= 1 – 15
* 14 * 13 = 0,265
28 27
26
5.
 ÿùèêå ñîäåðæèòñÿ 6 äåòàëåé, èçãîòîâëåííûõ íà 1-ì
çàâîäå, 2 äåòàëè íà 2-ì çàâîäå
è 4 äåòàëè íà 3-ì çàâîäå. Âåðîÿòíîñòü áðàêà íà
çàâîäàõ ðàâíà 19 , 19 è 59
20 50 100
Íàéòè âåðîÿòíîñòü òîãî, ÷òî íàóäà÷ó
èçâëå÷åííàÿ äåòàëü áóäåò êà÷åñòâåííàÿ.
Í1 – äåòàëü ñ 1-ãî çàâîäà
Í2 - äåòàëü ñî 2-ãî çàâîäà
Í3 - äåòàëü ñ 3-ãî çàâîäà.
Ð(Í1) = 6 = 1 ; Ð(Í2) = 2 = 1 ; Ð(Í3)
= 4 = 1
12 2 12 6
12 3
À - èçâëå÷åííàÿ äåòàëü êà÷åñòâåííàÿ
_
_ _ _
Ð (À) = Ð *(À) * Ð (Í1) + Ð *(À) * Ð (Í2) + Ð *(À)*Ð (Í3) =19 * 1 +
19 * 1 + 59 *1=147=>
Í1
_ Í2 Í3
20 2 50 6 100 3 200
Ð (À) = 1 – Ð (À) = 53/200
__________________________________________________________________________________________
6.Íåçàâèñèìûå âåðîÿòíûå âåëè÷èíû Õ,Ó ïðåäñòàâëÿþò
òîëüêî öåëûå çíà÷åíèÿ
Õ: îò 1 äî 16 ñ âåð-þ 1
16
Ó: îò 1 äî 23 ñ âåð-þ 1
23
Ð ( Õ+Ó = 32)
Õ Ó Ð (Õ=9; Õ =23) = P (Õ=9) * Ð (Ó = 23) = 1 * 1
9 23
16 23
10 22
P
( X+y=32 )=P ( X=8, y=23 ) + P ( X=10; y=12
)+…+P ( y=16,X=16 )=
16 16 = 8* 1 * 1
= 1
16 23
46
_________________________________________________________________________________________
7.Íåçàâèñèìûå ñëó÷àéíûå âåëè÷èíû Õ , Ó ïðèíèìàåò
òîëüêî öåëûå çíà÷åíèÿ.
Õ: îò 1 äî 14 ñ
âåðîÿòíîñòüþ 1
14
Ó: îò 1 äî
7 ñ âåðîÿòíîñòüþ 1
7
Íàéòè âåðîÿòíîñòü òîãî, ÷òî Ð (Õ £ Ó)
Åñëè Ó = 7, òî 1 £ Õ £ 6 1
* 6
7
14
Åñëè Ó = 6 òî 1£ Õ £ 5 1 *
5
7 14
Åñëè Ó = 5 òî 1£ Õ £ 4 1 * 4
7
14
Åñëè Ó = 4 òî 1£ Õ £ 3 1 * 3
7
14
Åñëè Ó = 3 òî 1£ Õ £ 2 1 * 2
7
14
Åñëè Ó = 2 òî 1 = Õ 1 * 1
7
14
Ð (Õ<Ó) = 1 * 6 + 1 * 5
+ 1 * 1 = 1+2+3+4+5+6 = 21 = 3
7 14 7 14
7 4 7 * 14 714 14
_________________________________________________________________________________________
8.Íåçàâèñèìûå âåëè÷èíû Õ1……Õ7 ïðèíèìàþò òîëüêî öåëûå çíà÷åíèÿ
îò
0 äî 10 ñ âåðîÿòíîñòüþ 1
11
Íàéòè âåðîÿòíîñòü òîãî , ÷òî Ð(Õ1…….Õ7) = 0
Ð (Õ1……Õ7 =0) = 1-Ð (Õ1….Õ7 ¹ 0) = 1- Ð( Õ1¹0….Õ7
¹
)=1-Ð( Õ1¹0 )*Ð (Õ2¹0)
7
*….* Ð(Õ7¹0) = 1 – 10 * 10 = 1 - 10
11……. 11 11
7 ðàç
9.Íåçàâèñèìûå ñëó÷àéíûå âåëè÷èíû Õ, Ó, Z ïðèíèìàþò öåëûå çíà÷åíèÿ
Õ: îò 1 äî 13 ñ âåðîÿòí-þ 1
13
Ó: îò 1 äî 12 _____/_____ 1
12
Z îò 1 äî 9 _____/_____ 1
9
Âåðîÿòíîñòü òîãî, ÷òî Õ;Ó;Z. ïðèìóò
ðàçíûå çíà÷åíèÿ?
Ïóñòü “Z” ïðèíÿëî êàêîå-òî çíà÷åíèå “à”. Ð
(Ó¹à)
= 11
12
Ïóñòü ïðè ýòîì Ó= â
Ð (Z ¹
a; Z ¹ â) = 11
; Ð = 11 * 11
13 12
13.
_______________________________________________________________________________________
10.
ì = Ì (Õ) - ? Ì (Õ) =
0,1+1,6+3,5 = 5,2
Ð ( Õ < ì) - ? Ð ( Õ < 5,2) = Ð(Õ=1) + Ð(Õ=4) =
0,5
___________________________________________________________________________________________
11.
Ä (Õ) - ?
Ì(Õ) = 0,4+0,9+2,5=3,8
2
Ì (Õ ) = 0,8+2,7+12,5 = 16
2 2 2
Ä (Õ) = Ì (Õ ) – Ì (Õ) = 16 - 3,8 = 1,56
______________________________________________________________________________________________________________
12. Íåçàâèñèìûå âåëè÷èíû Õ1,…….,Õ9 ïðèíèìàþò öåëîå çíà÷åíèå
– 8, - 7,…..,5,6
ñ âåðîÿòíîñòüþ 1
15
9
Íàéòè Ì (Õ1,Õ2,…..,Õ9) * Ì (Õ2,….,Õ9) = Ì (Õ1) * Ì(Õ2)*….* Ì(Õ9) =Ì (Õ9)
Ì (Õ1) = -8 * 1 – 7 * 1 * 6 * 1 - … + 5 * 1 + 6 * 1 = 1
(-8-7-5….+5+6) = -1
15 15
15 15 15 15
9 9
= Ì (Õ1) = ( -1) = -1
13.
Õ
|
8
|
10
|
12
|
14
|
16
|
Ð
|
0,25
|
0,2
|
0,2
|
0,2
|
0,25
|
ì= Ì
(Õ)-? Ì (Õ) = 2 + 2 + 1,2 + 2,8 + 4 = 12
ä(Õ)
-? 2 2
Ð ( (Õ-ì) < d) Ä
(Õ) = Ì (Õ – Ì (Õ) )
= Ì (Õ-12)
Õ-12
|
-4
|
-2
|
0
|
2
|
4
|
Ð
|
0,25
|
0,2
|
0,1
|
0,2
|
0,25
|
2
(Õ-12)
|
1
|
4
|
0
|
Ð
|
0,5
|
0,4
|
0,1
|
2
Ì (Õ-Ð) = 8+1,6
_____
d (Õ) = Ö d (Õ) » 3,1
Ð ( Õ –12 < 3,1 ) = Ð (-3,1<Õ –12 < 3,1) = Ð (8,9<Õ<15,1) =
= Ð (Õ=10) + Ð (Õ=12) + Ð (Õ=14) = 0,5
___________________________________________________________________________________________________________
14. Õ, Ó – íåèçâåñòíûå ñëó÷àéíûå âåëè÷èíû
Ì (Õ) =
3 8 2 2
2 2 2
Ì (Ó) =2 ½ Ä(ÕÓ) =
Ì( ÕÓ ) – Ì (ÕÓ) = Ì (Õ ) * Ì (Ó ) – [ Ì (Õ)*Ì (Õ)] =
Ä(Õ) = 4 ½
2 2
2 2
Ä(Ó) = 8 ½ Ä (Õ)=Ì(Õ ) –
Ì (Õ) = Ì (Õ ) = Ä (Õ) + Ì (Õ) = 4 + 9 = 13
Ä (Õ Ó) 2 2
Ì (Ó ) = Ä (Õ)
+ Ì (Ó) = 8 + 4 = 12
2
= 12*13 – (2 * 3) = 156 – 36 = 120
__________________________________________________________________________
15. Õ, Ó – íåçàâèñèìûå íåèçâåñòíûå âåëè÷èíû.
Ïðèíèìàþò çíà÷åíèå 0 è 1.
Ð (Õ=0) = 0,3 ½ 2 2
2 2 2
Ð (Ó=0) = 0,6 ½ Ì(Õ+Ó) + Ì (Õ + 2õó +ó ) = Ì
(Õ ) +2Ì (Õ) * Ì (Ó) + Ì (Ó ) =
2
Ì (Õ+Ó)
2
Ì (Õ) = 0,7 = Ì (Õ )
2
Ì (Ó) = 0,4 = Ì ( Ó )
= 0,7 + 2 * 0,7 * 0,4 + 0,4 = 1,66
16. Õ, Ó íåçàâèñèìûå íåèçâåñòíûå âåëè÷èíû
Ïðèíèìàþò çíà÷åíèå 0 è 1.
(çàäàíèå êàê â 15).
õ - ó
Ì (3 ) - ?
õ-ó õ -ó õ -ó
Ì (3 ) = Ì (3 * 3 ) =Ì (3 ) * Ì (3 ) = 2,4 * 2 = 1,6
3
Õ
-ó
Ì (3 ) = 0,3 + 2,1 =
2,4 Ì (3 )
= 0,5 + 0,5 = 4 * 0,5 = 1
3 3 3
_____________________________________________________________________________________________________________
17. Ïðîèçâîäèòñÿ 10240 íåçàâèñèìûõ èñïûòàíèé,
ñîñòîÿùèõ â òîì, ÷òî
ïîäáðàñûâàþòñÿ 9 ìîíåò
Õ – ÷èñëî èñïûòàíèé, â êîòîðûõ âûïàëî 3 ãåðáà
Ì (Õ) -?
1-èñïò. - 9 ìîíåò
9
èñïûòàíèé Ð = 1
2
3 3 6
3 9
Ð(Ã = 3) = Ñ9 * ( 1 ) * ( 1
) = Ñ9 *
( 1 ) =
84 * 1 - 21 = …
2
2 2 512 128
n = 10240 èñïûòàíèé
Ð = 21 ; Ì (Õ) = np = 21 * 10240 = 1680
128
128
18. Â ñåðèè íåçàâèñèìûõ èñïûòàíèé (îäíî
èñïûòàíèå çà åä.âðåìåíè)
âåðîÿòíîñòü íàñòóïëåíèÿ À ðàâíà 1
8.
Ïóñòü Ò-âðåìÿ îæèäàíèÿ íàñòóïëåíèÿ ñîáûòèÿ À
14 ðàç. Íàéòè Ì (Ò)1 Ä (Ò).
Õ1
– âðåìÿ îæèäàíèÿ äî ïåðâîãî íàñòóïëåíèÿ À
Õ2
– âðåìÿ îæèäàíèÿ îò ïåðâîãî íàñòóïëåíèÿ À äî 2-ãî
Ò = Õ1 + Õ2 +Õ3
+ …..Õ14
Õi Ð = 1
8 7/8
Ì (Õi) = 1
= 8 ; d = 7 Ä (Õi)
= d = = 56
8 8 2 2
p 1/8
Ì (Ò) = 14Ì * (Õ1) 14 * 8 = 112
Ä (Ò) = Ä(X1 ) = 14 * 56 = 784
19.
Âåëè÷èíû Õ1 …..Õ320 ðàñïðåäåëåíû ïî Áèíîìèíàëüíîìó çàêîíó ñ ïàðàìåòðàìè
ï
=4, ð = 3 Íàéòè Ì (Õ1 + Õ2 + …+ Õ320)=?
8
2 2 2
Ì (Õ1 +
…..+Õ 320) = 320Ì (Õ1 ) = Õ1 – áèíîìèíàëüíîå
2
2 Ì (Õ1) =
ïð = 3
=
Ì(Õ1 ) = Ä(Õ1) + Ì (Õ1)
= 2
2
Ä (Õ1 ) = nðq = 3 * 5
= 5
=
15 + 3 = 15 + 9 = 51
2 8 16
16 2 16 4 16
= 320 * 51 = 1020
16
_____________________________________________________________________________________________________________________
20.
Âåëè÷èíû Õ1 …..Õ18 ðàñïðåäåëåíû ïî çàêîíó Ïóàññîíà ñ îäèíàêîâûì
ìàò.
îæèäàíèÿì ðàâíûì 8.
2 2
Íàéòè
Ì (Õ1 +…+ Õ18 ) - ?
M (Õ) = Ä (Õ) = l = 8
2
2 2 2
Ì
(Õ1 +…+ Õ18 ) = 18 Ì (Õ1
) = 18 (Ä (Õ1) + Ì (Õi )
)=18(8 + 64)=18 * 72=1296
_________________________________________________________________________________________________________
21.
Õ – ðàâíîìåðíî ðàñïðåäåë¸í íà îòð. [ - 8,2 ]
Ð ( 1 )>5 = Ð (0< Õ <1 ) = > (0< Õ <0,5) =
Õ 5
1 – 5 >0 ; 1 – 5Õ > 0; Õ –1/5 < 0 Û (0< Õ <0,5)
Õ
Õ Õ
1
– 5Õ > 0; Õ – 1/5 < 0
Õ Õ
[ õ, â ]
0,Õ>à
0; Õ <à
f (Õ)= 1 ; à < Õ < â F (Õ) = õ – à ; à £ Õ £ à Û 0< Õ 1/5
â –î â –à
0,Õ >
â 1, Õ >B
F (Õ) = Õ + 8 = F (1/5) - F ( 0 ) =1/5 + 8 - 8
= 1
5 10 10 50
_______________________________________________________________________________________________________________________
22.
Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ -17; 10 ]
2 2
Ð
( Õ > 64) = 1- Ð ( Õ < 64) = 1 – 16
27
2
Ð (Õ < 64 ) = Ð (-8 < Õ <8) =
0; Õ < -17
F(Õ) = Õ + 17 , -17 £ Õ £ 10
27
1, Õ > 10
=
F (8) – F (-8) = 8 + 17 - -8 +
17 = 16
27 27 27
______________________________________________________________________________________________________________
23. Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ -1; 1 ]
8/9 X [a,b] ; f (x)
Ì (
Õ ) a
0; x <-1
M(x)= ∫ x f(x) dx f (x)= -1<x<1
b 0; x>1
a
M(y(x))=∫
y
(x) f (x) dx
b
8/9 1 8/9 17/9 1
M(X
) = ∫ ½* X DX = ½ * X
= 9/17
-1 17/9 -1
24. Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ 0.1 ]
9/10 9/10
Ä (
19Õ ) = 361 (Õ )
9/10 9/10 2
2 9/10 9/4 2 9/10 9/10 * 2
Ä
(Õ ) = Ì ( (Õ ) ) - Ì (Õ ) =
Ì (Õ ) - Ì (Õ ) Õ
__________________________________________________________________________________________________________
25.
Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ 5; 8 ] * Ä (24x+ 36) - ?
Ä (24Õ +
36) = Ä (24Õ) = 576 * Ä (Õ) = 576 * 3 = 432
2
4
Ä (Õ) = (
â – à )
12
2
Ä (Õ) = 8
– 5 = 9 = 3
12
12 4
_______________________________________________________________________________________________________________
26. Õ1,……Õ2 – Íåçàâèñèìûå è ðàñïðåäåëåííûå ïî
ïîêàçàòåëüíîìó çàêîíó.
2
Íàéòè Ì
[ (Õ1 + Õ2 + …..+ Õ10) ], åñëè Ì (Õi
) = 4.
Ì (Õ) =
1
l
Ä (Õ)
= 1
2
l
M (Õi ) = > Ä (Õi) = 16
2
2 2
Ì
[ (Õ1 +….+ Õ10) ]=Ä(Õ1 +…+ Õ10) + Ì (Õ1 +….+ Õ10) =10Ä (Õ1)+[ 10Ì (Õ1) ]=
2
=
160 + ( 10 * 4) = 1760
_________________________________________________________________________________________________________________
2
Ì(Õ) =1/ l ; Ä(Õ) = 1/l
27.
Õ –ðàñïðåäåëåí ïî ïîêàçàòåëüíîìó
ïðèçíàêó
2
Íàéòè
Ì [ (Õ + 8) ] , åñëè Ä (Õ) = 36 Ì (Õ)=6
2
2 2
2
Ì
(Õ + 8) = M(Õ + 16õ + 64) = Ì (Õ )
+ 16Ì (Õ) + Ì (64) = Ä (Õ) + Ì (Õ) +
+ 16 Ì(Õ) + 64 =36 + 36 + 96
+ 64 =232
____________________________________________________________________________________________________________
28.
Õ –ïîêàçàòåëüíîå ðàñïðåäåëåíèå; Õ – ïîêàçàòåëüíûé çàêîí
0, Õ < 0
F (Õ) = -2õ
1 – å , Õ >0, Íàéòè Ln (1
– Ð ( Õ < 6) ) = Ln (1 – F (6) ) =
-6/7
-6/7 -6/7
= F (6)
= 1 – å = Ln ( 1 – (1 – å ) ) = Ln å = - 6/7
29. (Õ)
- ñëó÷àéíàÿ âåëè÷èíà
0, Õ < 10
ƒ
(Õ) = Ñ ; Õ ≥ 10
5
Õ
Ñ - ? ; Ì (Õ) - ?
¥ ¥ îïð.
B ¥
-5
∫ ƒ (Õ)dõ = 1 => ∫ ñ dõ = lim ∫ = cdx = C lim ∫ X dx =
10
10 5 b->¥ 10
5 b->¥ 10
Õ X
b
-4 -4 4
4 4
= C * lim X = C lim - b + 10
= C * 10 = >
1 = C 10 = >
b->¥ -4 b->¥ 4 4
4 4
10
4
=> C = 4 * 10
0; Õ < 10
ƒ
(Õ) = 4
4 * 10 , Õ ³ 10
5
Õ
¥ ¥ 4
Ì (Õ)
= ∫
Õ ƒ (Õ) dx = ∫
4 * 10 dx
10 10 4
Õ
_________________________________________________________________________________
30. Õ –
íîðìàëüíàÿ ñëó÷àéíàÿ âåëè÷èíà
Ì (Õ) =
16
Ä
(Õ) = 25
? – Ð (Õ>10,5)
= 1
- f 10,5 – 16 = 0,5 + f (1,1) = 0,5 + 0,364 = 0,864
2
5
________________________________________________________________________________________
1.Ð (d £ X £ b ) = f b – m - f d - m
d d
2.
P ( X < b ) = 1 + f b – m
2
d
3.
P ( X > b ) = 1 - f b – m
2 d