Примеры решённых задач по теории вероятности

G_ihkj_^kl\_ggucih^kqzl\_jhylghkl_cG_aZ\bkbfhklvkh[ulbcL_hj_fukeh`_gbybmfgh`_gby\_jhylghkl_c1.1.GZclb\_jhylghklvpnlh]hqlhkemqZcgh\u[jZggh_gZlmjZevgh_qbkehbafgh`_kl\Z{1, 2,…, n} ^_eblkygZnbdkbjh\Zggh_gZlmjZevgh_qbkehk. GZclb lim pn .n →∞nJ_r_gb_kj_^baZ^Zggh]hgZ[hjZqbk_ebf__lky   djZlguokIhwlhfmihdeZkkbq_k n n 1  n −1kdhfmhij_^_e_gbx\_jhylghklb pn =  k  ⇒ lim pn = ≤ pn ≤.→∞nnk  nknk 1.2. Ba fgh`_kl\Z ^ « n` kemqZcgh \u[bjZ_lky qbkeh a GZclb \_jhylghklv lh]hqlhqbkehag_^_eblkygbgZh^ghbagZlmjZevguobihiZjgh\aZbfghijhkluoqbk_ea1,…,akGZclb lim pn .n →∞nJ_r_gb_gZdZ`^h_baqbk_eai ^_eblky   qbk_ebaZ^Zggh]hgZ[hjZlh]^Zihkdhev ai kndmai ihiZjgh\aZbfghijhklugbgZh^ghbagbog_^_eylky xn = n − ∑   qbk_eIhdeZki =1  ai kbq_kdhfmhij_^_e_gbx\_jhylghklb pn =kxn1; lim pn = 1 − ∑ ZgZeh]bqghn n →∞i =1 ai1.3.

Bafgh`_kl\Z^«n`kemqZcgh\u[bjZ_lkyqbkehaImklvpn – \_jhylghklvlh]hqlha2 – 1 ^_eblkygZGZclb lim pn .n →∞J_r_gb_bkdhfh_kh[ulb_j_Zebam_lky\ljzokemqZyo±Z±djZlghZdjZlghZ±djZlghbZdjZlghIhke_^gbckemqZck\h^blkyd^\mfi_j\ufihkdhevdm_kebh^ghbaqbk_e±qzlgh_lhb^jm]h_hlebqZxs__kygZlZd`_ qzlgh_ Ih1 n − 1  n + 1p;lim.wlhfm pn = +=n5 10   10  n →∞1.4.Badheh^udZjldZjlu\ugbfZxlh^gh\j_f_gghdZjluJZkkfZljb\Zxlkykh[ulby:±kj_^b\ugmluodZjlohly[uh^gZ[m[gh\Zy<±kj_^b\ugmluodZjlohly[uh^gZq_j\hggZyGZclb\_jhylghklvkh[ulby C = A ∪ B.4n44313 1J_r_gb_: p A = pB = 1 −   ; p A∩ B = 1 −   ⇒ pC = p A + pB − p A∩ B = 1 − 2   +   .424 21.5.ImklvΩ = {w1, w2, w3, w4} – ijhkljZgkl\hwe_f_glZjguobkoh^h\b\k_we_f_glZjgu_ bkoh^u jZ\gh\_jhylgu JZkkfZljb\Zxlky ke_^mxsb_ kh[ulby A = {w1, w2}; B = {w1,w3}; C = {w1, w4}. GZclb\_jhylghklbkh[ulbc:, <bKY\eyxlkyebmdZaZggu_kh[ulbyg_aZ\bkbfufb"A 1J_r_gb_ kh]eZkgh deZkkbq_kdhc ko_f_ jZkqzlZ \_jhylghklb p ( A) == = p (B) =Ω 2= p (C ).

Kh[ulby:b<y\eyxlkyg_aZ\bkbfufb_kebp(AB) = p(A)·p(B). AB = BC = AC = :<K= {w1} ⇒ p ( AB ) =1 1 1= ⋅ = p ( BC ) = p ( AC ) = p ( ABC ) , ihwlhfm iZju kh[ulbc :<, :K b4 2 23<Kg_aZ\bkbfuH^gZdhkh[ulby:, <bKaZ\bkbfu\kh\hdmighklbihkdhevdm1 1≠  .4 21.6. Imklv j_amevlZl g_dhlhjh]h kemqZcgh]h wdki_jbf_glZ khklhbl \ ihy\e_gbb ljzof_jgh]h^\hbqgh]h\_dlhjZo1, o2, o3ijbqzf\_jhylghklbj1,... j8ihy\e_gby\_dlhjh\khhl\_lkl\_gghlZdh\up1 = P {( 0, 0, 0 )} = 0,14;p2 = P {(0, 0,1)} = 0,11;p3 = P {( 0,1, 0 )} = 0,11;p4 = P {(0,1,1)} = 0,14;p5 = P {(1, 0, 0 )} = 0,11;.p6 = P {(1, 0,1)} = 0,14;p7 = P {(1,1, 0 )} = 0,14;p8 = P {(1,1,1)} = 0,11.JZkkfZljb\Zxlkykh[ulby: = {(x1, x2, x3): x1 = 0}, B = {(x1, x2, x3): x2 = 0}, C = {(x1, x2,x3): x3 = 0}.

GZclb\_jhylghklbkh[ulbc:, <bKY\eyxlkyebmdZaZggu_kh[ulbyg_aZ\bkbfufb"J_r_gb_ kh[ulby b y\eyxlky g_i_j_dju\Zxsbfbky qZklyfb kh[ulby : ihwlhfmp(A) = p1 + p2 + p3 + p4 = 0.5; ZgZeh]bqghp(B) = p(C) = 0.5. p(AB) = p1 + p2 = 0.25; ZgZeh]bqghp(AC) = p(BC) = 0.25 = p(A)·p(B). LZdbfh[jZahfkh[ulbyA, B bKihiZjghg_aZ\bkbfubaZ\bkbfu\kh\hdmighklb1.7.;jhkZxlky^\_b]jZevgu_dhklbImklvj±qbkehhqdh\\uiZ\r__gZi_j\hcdhklbk±qbkehhqdh\\uiZ\r__gZ\lhjhcdhklbJZkkfZljb\Zxlkyke_^mxsb_kh[ulby: = {(j,k):k = 1,2,5}; B = {(j,k): k = 4,5,6}, C = {(j,k): j + k =9}. GZclb\_jhylghklbkh[ulbc:, <, K, :<,:K, <K, :<K.A 3⋅ 6 1J_r_gb_ kh]eZkgh deZkkbq_kdhc ko_f_ jZkqzlZ \_jhylghklb p ( A) === =Ω 6⋅6 2= p ( B ) , ]^_Ω±ijhkljZgkl\h\k_o\hafh`guobkoh^h\\ZjbZglh\\uiZ^_gby^\modhk1l_cK = {(4,5); (5,4); (3,6); (6,3)} ⇒ p (C ) = . AB = {(j,k): k = 5}, AC = {(4,5)}, BC = {(4,5);9111(5,4); (6,3)}, ABC = {(5,4)} ⇒ p ( AB ) = , p ( AC ) = p ( ABC ) = , p ( BC ) = .63612<aZ^ZqZo±lj_[m_lkyhij_^_eblvgZ^z`ghklv\_jhylghklv[_ahldZaghcjZ[hlu\l_q_gb_g_dhlhjh]hnbdkbjh\Zggh]h\j_f_gbL ijb[hjZ ko_fZ dhlhjh]h ijb\_^_gZ \ khhl\_lkl\mxs_c aZ^Zq_ GZ^z`ghklv dZ`^h]h we_f_glZ jZ\gZ j Ijb[hj kqblZ_lky jZ[hlZxsbf _keb g_ ih\j_`^_gghc hklZzlky ohly [u h^gZ p_ihqdZ hl ©\oh^Z d \uoh^mª ko_fuIj_^iheZ]Z_lkyqlhwe_f_glu\uoh^ylbakljhyg_aZ\bkbfh^jm]hl^jm]Z1.8.J_r_gb_ ijb ihke_^h\Zl_evghf kh_^bg_gbb g_h[oh^bfZ h^gh\j_f_ggZy jZ[hlZ \k_owe_f_glh\ lh _klv gZ^z`ghklb i_j_fgh`Zxlky ijb iZjZee_evghf kh_^bg_gbb g_h[oh^bfZjZ[hlZohly[uh^gh]hbawe_f_glh\lh_klvgZ^z`ghklvfh`_l[ulvjZkkqblZgZdZd_^bgbpZfbgmk\_jhylghklv©hldexq_gbyª\k_owe_f_glh\iZjZee_evghcp_ib±qn, ]^_q = 1 – p.

<^ZgghfkemqZ_ P = 1 − q 1 − p 2 (1 − q 2 ) .()1.9.()J_r_gb_ZgZeh]bqgh P = 1 − (1 − p 2 ) 1 − p (1 − q 2 ) .1.10.( (J_r_gb_ZgZeh]bqgh P = 1 − q 1 − 1 − (1 − p 2 )2)) = 1 − q (1 − p ) .2 21.11.( ())J_r_gb_ZgZeh]bqgh P = 1 − (1 − p 2 ) 1 − 1 − q (1 − p 2 ) = 1 − q (1 − p 2 ) .1.12.( ())J_r_gb_ZgZeh]bqgh P = 1 − q 1 − 1 − q (1 − p 3 ) = 1 − q 2 (1 − p 3 ).1.13.()J_r_gb_ZgZeh]bqgh P = 1 − q 1 − (1 − q 2 ) .21.14.J_r_gb_ZgZeh]bqgh P = p 2 (1 − q 2 ) .221.15.J_r_gb_ZgZeh]bqgh P = (1 − q 2 )(1 − q 3 ) .21.16.J_r_gb_ZgZeh]bqgh P = p 2 (1 − q 2 )(1 − q 3 ) .1.17.()J_r_gb_ZgZeh]bqgh P = p (1 − q 3 ) 1 − q (1 − p 2 ) .NhjfmeZiheghc\_jhylghklb2.1.Bf_xlky^\_mjgu<i_j\hcmjg_Z1[_euorZjh\bb1±qzjguoZ\h\lhjhcmjg_Z2[_euorZjh\bb2±qzjguoBadZ`^hcmjguba\e_dZxlihh^ghfmrZjmbi_j_deZ^u\Zxlbo\^jm]mxmjgmbai_j\hc\h\lhjmxba\lhjhc±\i_j\mxAZl_fbai_j\hcmjguba\e_dZxl[_a\ha\jZs_gbynrZjh\DZdh\Z\_jhylghklvqlhkj_^bgbohdZ`_lkym[_euo"4J_r_gb_ ih nhjfme_ iheghc \_jhylghklb p = ∑ p ( Ai ) p ( Bi ), ]^_ – ba\e_q_gb_ gZi =1i_j\hfwlZi_bamjg^\mo[_euorZjh\±[_eh]hbqzjgh]h±qzjgh]hb[_eh]h±^\moqzjguoAi – \_jhylghklvkhhl\_lkl\mxs_]hbkoh^ZgZi_j\hfwlZi_<i±gZ\lhjhfIhwlhfmCam1 ⋅ Cbn1− mCam1 −1 ⋅ Cbn1−+m1Cam1 +1 ⋅ Cbn1−−1ma1a2a1b2b1a2p=+++( a1 + b1 )( a2 + b2 ) Can1 +b1( a1 + b1 )( a2 + b2 ) Can1 +b1( a1 + b1 )( a2 + b2 ) Can1 +b1Cam1 ⋅ Cbn1− mb1b2+.( a1 + b1 )( a2 + b2 ) Can1 +b12.2.Bamjgu\dhlhjhcgZoh^ylkyZ1[_euorZjh\bb1qzjguom^Zeyxlh^bgrZjIhke_wlh]hbamjguba\e_dZxlihhq_jz^ghk\ha\jZs_gb_fnrZjh\DZdh\Z\_jhylghklvqlhkj_^bgbo[m^_lm[_euorZjh\"mn−ma1 Ca1 + m − 2 ⋅ Cb1 + n − m −1J_r_gb_ ih nhjfme_ iheghc \_jhylghklb ihemqbf p =+a1 + b1Can1 +b1 + n− 2mn−mb1 Ca1 + m −1 ⋅ Cb1 + n− m − 2+.a1 + b1Can1 +b1 + n− 22.3.Bf_xlky^\_mjgu<i_j\hcmjg_Z1[_euorZjh\bb1±qzjguo\h\lhjhcmjg_Z2[_euobb2qzjguoBadZ`^hcmjgum^Zeyxlihh^ghfmrZjmAZl_f\k_rZjuba\lhjhcmjgui_j_deZ^u\Zxl\i_j\mxbbag_zba\e_dZxl[_a\ha\jZs_gbynrZjh\DZdh\Z\_jhylghklvqlhkj_^bgbohdZ`_lkym [_euo"J_r_gb_ ih nhjfme_ iheghc \_jhylghklb+p=Cam1 + a2 − 2 ⋅ Cbn1−+mb2a1a2+( a1 + b1 )( a2 + b2 ) Can1 + a2 +b1 +b2 −2Cam1 + a2 −1 ⋅ Cbn1−+mb2 −1Cam1 + a2 −1 ⋅ Cbn1−+mb2 −1Cam1 + a2 ⋅ Cbn1−+mb2 − 2a1b2a2b1b1b2.++( a1 + b1 )( a2 + b2 ) Can1 + a2 +b1 +b2 − 2 (a1 + b1 )( a2 + b2 ) Can1 + a2 +b1 +b2 −2 ( a1 + b1 )(a2 + b2 ) Can1 + a2 +b1 +b2 −22.4.Bf_xlky^\_mjgu<i_j\hcmjg_Z1[_euorZjh\bb1qzjguo\h\lhjhcmjg_a2[_euorZjh\bb2qzjguoBai_j\hcmjguba\e_dZxlrZjZbi_j_deZ^u\Zxlbo\h\lhjmxmjgmAZl_fba\lhjhcmjguba\e_dZxlihhq_j_^ghk\ha\jZs_gb_fnrZjh\DZdh\Z\_jhylghklvqlhkj_^bgbohdZ`_lkym[_euorZjh\"Ca21 Cam2 + m +1 ⋅ Cbn2−+mn − m −1 a1b1 Cam2 + m ⋅ Cbn2−+mn − mCb21 Cam2 + m −1 ⋅ Cbn2 + n − m +1J_r_gb_: p = 2.+ 2+ 2Ca1 +b1Can2 +b2 + n+1Ca1 +b1 Can2 + b2 + n+1Ca1 +b1Can2 +b2 + n+12.5.

Bf_xlky^\_mjgu<i_j\hcmjg_Z1[_euorZjh\bb1qzjguo\h\lhjhcmjg_a2[_euorZjh\bb2qzjguo\lj_lv_cmjg_a3[_euorZjh\bb3qzjguoBai_j\uo^\mo mjgba\e_dZxlihh^ghfmrZjmbi_j_deZ^u\Zxlbo\lj_lvxmjgmAZl_fbalj_lv_cmjguba\e_dZxl[_a\ha\jZs_gbynrZjh\DZdh\Z\_jhylghklvqlhkj_^bgbohdZ`_lkym[_euorZjh\"J_r_gb_:Cam3 + 2 ⋅ Cbn3− mCam3 +1 ⋅ Cbn3−+m1Cam3 ⋅ Cbn3−+m2a1a2a1b2 + a2b1b1b2p=++.( a1 + b1 )( a2 + b2 ) Can3 +b3 +2 ( a1 + b1 )( a2 + b2 ) Can3 +b3 +2 ( a1 + b1 )( a2 + b2 ) Can3 +b3 +22.6. < mjg_ gZoh^blky [_euo rZjh\ b qzjguo AZl_f \ mjgm ^h[Z\ey_lky rZjZijbqzfdZ`^uc^h[Z\ey_fucrZjg_aZ\bkbfhhlhklZevguok\_jhylghklvxphdjZrb\Z_lky\[_eucp\_lbk\_jhylghklvxq = 1 – p \qzjgucp\_lIhke_wlh]hbamjguba\e_dZxl[_a\ha\jZs_gbyrZjh\DZdh\Z\_jhylghklvqlhkj_^bgbohdZ`_lky[_euo"C7 ⋅C3C 7 C3C7 C3J_r_gb_ ih nhjfme_ iheghc \_jhylghklb p = p3 12 10 8 + p 2 q 1110 9 + pq 2 10 1010 +C20C20C20+ q3C97 C113.10C202.7.

<mjg_gZoh^blky[_euorZjh\bqzjguoAZl_fijhba\h^blkyqZklbqgZyaZf_gZrZjh\ke_^mxsbfh[jZahfdZ`^uc[_eucrZjg_aZ\bkbfhhl^jm]bok\_jhylghklvxphklZ\ey_lkygZf_kl_Zk\_jhylghklvxq = 1 – p hdjZrb\Z_lky\qzjgucp\_lIhke_wlh]hbamjgu ba\e_dZxl ihhq_jz^gh k \ha\jZs_gb_f rZjZ DZdh\Z \_jhylghklv qlh kj_^b gbohdZ`_lkyohly[uh^bg[_eucrZj"J_r_gb_Bkdhfh_kh[ulb_h[jZlghdba\e_q_gbxljzoqzjguorZjh\lh]^Zihnhjfm35k 5 − k C5 + k + 3−1 e_iheghc\_jhylghklb p = ∑ 1 − p q.C103 + 3−1 k =0 2.8. < mjg_ gZoh^blky [_euo rZjh\ b qzjguo DZ`^uc [_euc rZj g_aZ\bkbfh hl^jm]bok\_jhylghklvxp m^Zey_lkyba mjguZk\_jhylghklvxq = 1 – phklZ\ey_lky \g_cIhke_ wlh]h ba mjgu ba\e_dZxl ihhq_jz^gh k \ha\jZs_gb_f rZjZ DZdh\Z \_jhylghklvqlhkj_^bgbohdZ`_lkyohly[uh^bg[_eucrZj"5C3J_r_gb_ZgZeh]bqgh p = ∑ 1 − p k q5− k 3 5+3−1 .C10− k + 3−1 k =0 KemqZcgu_\_ebqbgu>bkdj_lgu_jZkij_^_e_gby[bghfbZevgh_]_hf_ljbq_kdh_IZkdZey3.1.Ijhba\h^ylNk_jbcih[jhkZgbcb]jZevghcdhklb\dZ`^hck_jbbDZdh\Z\_jhylghklvqlhjh\gh\kk_jbyo≤ k ≤ Nr_klvhqdh\\uiZ^_l[he__ljzojZa"J_r_gb_j1±\_jhylghklvlh]hqlhr_klvhqdh\\uiZ^_l[he__ljzojZaohly[u\h^10ghck_jbb3khklZ\ey_l92815 1 55p1 = 1 − q10 − pq 9 − p 2 q 8 − p 3q 7 = 1 −   −   −     −66 6 6671 5−     , ]^_±ihegZy\_jhylghklvZ\uqblZ_fu_±\_jhylghklb\uiZ^_gbyr_klbhq6 6dh\bebjZaZjZkkqblZggu_^ey^bkdj_lgh]hjZ\ghf_jgh]hjZkij_^_e_gby<_jhylghklv g_h[oh^bfh]h bkoh^Z jh\gh \ k k_jbyo fh`_l [ulv jZkkqblZgZ k ihfhsvx nhjfmeuN −k^ey[bghfbZevgh]hjZkij_^_e_gby P = C Nk p1k (1 − p1 ) .3.2.Ijhba\h^ylN g_aZ\bkbfuobkiulZgbcBkiulZgb_khklhbl\[jhkZgbyob]jZevghcdhklb^h\lhjh]hihy\e_gbyqbkeZhqdh\djZlgh]hljzfmki_ohfkqblZ_lky\lhjh_ihy\e_gb_mdZaZggh]h qbkeZ hqdh\ g_ iha`_ iylh]h [jhkZgby dhklb K dZdhc \_jhylghklvx ohly [u \h^ghfbaN g_aZ\bkbfuobkiulZgbcijhbahc^zlmki_o"J_r_gb_: p1±\_jhylghklvg_h[oh^bfh]hbkoh^Z\h^ghfbkiulZgbb_zfh`ghjZkkqb1lZlv ih nhjfme_ ^ey jZkij_^_e_gby IZkdZey  p =  .