Теория вероятностей и математическая статистика. Билялов Р.Ф. - 11 стр.

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C
5
15
= 13 · 11 · 7 · 3. |A| = C
5
7
= 21, P (A) = C
5
7
/C
5
15
= 1/143,
|B| = C
3
3
· C
2
12
= 66,
P (B) =
66
C
5
15
= 2/91, P (C) =
C
5
10
C
5
15
= 12/143.
A B
ω = (i
1
, i
2
, i
3
), 1 ≤ i
k
≤ 5, i
1
6= i
2
6= i
3
|Ω| = 5 · 4 · 3 = 60 P (A) = 1/60 |B| = 4 · 3 · 2 = 24
P (B) = 24/60 = 0.4.
A
¯
A, B A
¯
B.
|Ω| = 6
2
= 36.
A A = {ω : ω = (1, i); i = 1, ..., 6}.
P (A) = |A|/|Ω| = 6/36 = 1/6 P (
¯
A) = 1 − P (A) = 5/6. B = {ω :
(6, i), (i, 6), (6, 6); i = 1, ..., 5} P (B) = 11/36 A
¯
B = {ω : (1, i) : i =
1, ..., 5) P (A
¯
B) = 5/36.
n
n! n P
n
= n!
|Ω| = n!.
A
(n −1)
|A| = (n − 1)! · 2.
P (A) = 2 · (n − 1)!/n! = 2/n.
   Ðåøåíèå. ×èñëî âñåâîçìîæíûõ âûáîðîâ 5 ñòóäåíòîâ èç 15 ðàâíî
      5
     C15 = 13 · 11 · 7 · 3. |A| = C75 = 21, P (A) = C75 /C15
                                                          5
                                                             = 1/143,

                           |B| = C33 · C12
                                        2
                                           = 66,
                        66                   5
                                            C10
              P (B) =    5  = 2/91, P (C) =  5 = 12/143.
                        C15                 C15
   Çàäà÷à 14.92 Íà ïÿòè êàðòî÷êàõ íàïèñàíû öèôðû îò 1 äî 5.
Îïûò ñîñòîèò â ñëó÷àéíîì âûáîðå òðåõ êàðòî÷åê è ðàñêëàäûâàíèè
èõ â ïîðÿäêå ïîñòóïëåíèÿ â ðÿä ñëåâà íàïðàâî. Íàéòè âåðîÿòíîñòü
ñîáûòèé: A={ïîÿâèòñÿ ÷èñëî 123}, B ={ïîÿâèòñÿ ÷èñëî, íå ñîäåðæà-
ùåå öèôðû 3}.
     Ðåøåíèå. ω = (i1 , i2 , i3 ), 1 ≤ ik ≤ 5, i1 6= i2 6= i3  óðíîâàÿ ñõåìà
áåç âîçâðàùåíèÿ, |Ω| = 5 · 4 · 3 = 60, P (A) = 1/60, |B| = 4 · 3 · 2 = 24,
P (B) = 24/60 = 0.4.
     Çàäà÷à 2.1. Áðîøåíû äâå èãðàëüíûå êîñòè. Ïðåäïîëàãàÿ, ÷òî
ýëåìåíòàðíûå ñîáûòèÿ ðàâíîâåðîÿòíû, íàéòè âåðîÿòíîñòü ñîáûòèé:
A={íà 1-é êîñòè âûïàëà 1}, Ā, B ={âûïàëà õîòÿ áû îäíà 6}, AB̄.
     Ðåøåíèå.|Ω| = 62 = 36. Âûïèøåì âñå ýëåìåíòàðíûå ñîáûòèÿ, èç
êîòîðûõ ñîñòîèò ñîáûòèå A: A = {ω : ω = (1, i); i = 1, ..., 6}. Òîãäà
P (A) = |A|/|Ω| = 6/36 = 1/6, P (Ā) = 1 − P (A) = 5/6. B = {ω :
(6, i), (i, 6), (6, 6); i = 1, ..., 5}, P (B) = 11/36. AB̄ = {ω : (1, i) : i =
1, ..., 5), P (AB̄) = 5/36.
     Çàäà÷à 2.2. Íà ïîëêå â ñëó÷àéíîì ïîðÿäêå ðàññòàâëåíî n êíèã,
ñðåäè êîòîðûõ íàõîäèòñÿ äâóõòîìíèê Ä. Ëîíäîíà. Ïðåäïîëàãàÿ, ÷òî
ðàçëè÷íûå ðàñïîëîæåíèÿ êíèã ðàâíîâåðîÿòíû, íàéòè âåðîÿòíîñòü
òîãî, ÷òî îáà òîìà äâóõòîìíèêà ðàñïîëîæåíû ðÿäîì.
     Ðåøåíèå. ×èñëî ðàçëè÷íûõ ñïîñîáîâ ðàññòàíîâêè n êíèã ðàâ-
íî n! (÷èñëî ïåðåñòàíîâîê èç n ýëåìåíòîâ ðàâíî Pn = n!). Èòàê,
|Ω| = n!. ×åìó ðàâíÿåòñÿ ÷èñëî ýëåìåíòàðíûõ ñîáûòèé, áëàãîïðè-
ÿòíûõ ñîáûòèþ A={ îáà òîìà äâóõòîìíèêà îêàæóòñÿ ðÿäîì}? Ðàñ-
ñìîòðèì îáà òîìà êàê îäíó êíèãó. Òîãäà áóäåò (n − 1) êíèãà. Ó÷òåì
òàêæå, ÷òî îáúåäèíåíèå äâóõ òîìîâ èìååò äâà âàðèàíòà (1-é òîì, 2-é
òîì) è (2- é òîì, 1-é òîì). Ñëåäîâàòåëüíî, |A| = (n − 1)! · 2. Îòñþäà
P (A) = 2 · (n − 1)!/n! = 2/n.


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