Теория вероятностей и математическая статистика. Билялов Р.Ф. - 21 стр.

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P (E) = 0.5, P (F ) = 0.4, P (D) = 0.36, P (EF ) = 0.2, P (ED) =
0.08, P (F D) = 0.1, P (EF D) = 0.06. E F
P (E)P (F )P (D) = 0.07 6= 0.06.
B = A =
P (A|B) =
P (AB)
P (B)
=
C
1
12
C
1
2
C
2
20
=
3 · 8 · 2
5 · 19
=
48
95
.
A =
|A| = 12, B = {(3, 3)}, |Ω| = 36, P (B|A) =
P (AB)/P (A) = P (B)/P (A) = 1/12.
(ξ
1
, ξ
2
)
{(x
1
, x
2
) : 0 ≤ x
1
≤ 1, 0 ≤ x
2
≤ 1}.
C
1
= (ξ
1
≤ 1/2), C
2
= (ξ
2
≤ 1/2), C
3
= ((ξ
1
− 1/2)(ξ
2
− 1/2) < 0).
C
1
, C
2
, C
3
C
1
, C
2
, C
3
C
1
C
2
C
3
P (C
1
) = P (C
2
) = P (C
3
) = 1/2,
P (C
1
C
2
) = P (C
1
C
3
) = P (C
2
C
3
) = 1/4,
P (C
1
C
2
C
3
) = 0 6= P (C
1
)P (C
2
)P (C
3
) = 1/8.
C
1
, C
2
, C
3
, C
1
C
2
, C
3
A
1
A
2
A
3
A
k
p
k
,
k = 1, 2, 3.
    à) P (E) = 0.5, P (F ) = 0.4, P (D) = 0.36, P (EF ) = 0.2, P (ED) =
0.08, P (F D) = 0.1, P (EF D) = 0.06. E è F íåçàâèñèìû.
    á) íåò, òàê êàê P (E)P (F )P (D) = 0.07 6= 0.06.
    Çàäà÷à 14.182.  ÿùèêå ëåæàò 12 êðàñíûõ, 8 çåëåíûõ è 10 ñèíèõ
øàðîâ. Íàóäà÷ó âûíèìàþòñÿ äâà øàðà. Íàéòè âåðîÿòíîñòü òîãî, ÷òî
áóäóò âûíóòû øàðû ðàçíîãî öâåòà, ïðè óñëîâèè, ÷òî íå âûíóò ñèíèé
øàð.
    Ðåøåíèå. B = {íå âûíóò ñèíèé øàð}, A ={øàðû ðàçíîãî öâåòà}.
                         P (AB)  C1 C1    3·8·2   48
             P (A|B) =          = 122 2 =        = .
                          P (B)   C20     5 · 19  95

    Çàäà÷à 3.1. Áðîøåíî äâå èãðàëüíûå êîñòè. Êàêîâà âåðîÿòíîñòü
òîãî, ÷òî âûïàëî äâà ðàçà 3, åñëè èçâåñòíî, ÷òî ñóììà âûïàâøèõ
î÷êîâ äåëèòñÿ íà 3?
    Ðåøåíèå. A ={ñóììà âûïàâøèõ î÷êîâ äåëèòñÿ íà 3}= {(1,2),
(1,5),(2,4), (3,3),(3,6),(4,5), äàëåå òå æå ïàðû, íî öèôðû çàïèñàíû
â îáðàòíîì ïîðÿäêå}. |A| = 12, B = {(3, 3)}, |Ω| = 36, P (B|A) =
P (AB)/P (A) = P (B)/P (A) = 1/12.
    Çàäà÷à 3.7. Ñëó÷àéíàÿ òî÷êà (ξ1 , ξ2 ) èìååò ðàâíîìåðíîå ðàñïðå-
äåëåíèå â êâàäðàòå {(x1 , x2 ) : 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 1}. Ïóñòü
C1 = (ξ1 ≤ 1/2), C2 = (ξ2 ≤ 1/2), C3 = ((ξ1 − 1/2)(ξ2 − 1/2) < 0). Ïî-
êàçàòü, ÷òî ñîáûòèÿ C1 , C2 , C3 ïîïàðíî íåçàâèñèìû. ßâëÿþòñÿ ëè
ñîáûòèÿ C1 , C2 , C3 âçàèìíî íåçàâèñèìûìè? Çàâèñèìû ëè ñîáûòèÿ
C1 C2 è C3 ?
    Ðåøåíèå.
                      P (C1 ) = P (C2 ) = P (C3 ) = 1/2,
               P (C1 C2 ) = P (C1 C3 ) = P (C2 C3 ) = 1/4,
             P (C1 C2 C3 ) = 0 6= P (C1 )P (C2 )P (C3 ) = 1/8.
Êàê ñîáûòèÿ C1 , C2 , C3 , òàê è ñîáûòèÿ C1 C2 , C3 çàâèñèìû.
   Çàäà÷à 3.10. Ýëåìåíòû A1 è A2 ýëåêòðè÷åñêîé öåïè ñîåäèíåíû
ïàðàëëåëüíî, à A3 ïðèñîåäèíåí ê íèì ïîñëåäîâàòåëüíî. Âåðîÿòíîñòü
âûõîäà èç ñòðîÿ çà äàííûé ïåðèîä âðåìåíè ýëåìåíòà Ak ðàâíà pk ,
k = 1, 2, 3. Ïðåäïîëàãàåòñÿ, ÷òî ýëåìåíòû âûõîäÿò èëè íå âûõîäÿò


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