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bdkbZμ26% izkf;drk dk fl¼kar% ifjp; ,oa mi;ksx

16 uksV
3 ls foHkkT; la[;k dh izkf;drk =
50
blh izdkj 5 ls foHkkT; la[;k gksxhA
5 ls foHkkT; la[;k (5, 10, 15, 20, 25, 30, 35, 40, 45, 50)
10
5 ls foHkkT; la[;k =
50
10
5 ls foHkkT; la[;k dh izkf;drk =
50
ijUrq 30 vkSj 45 nks la[;k,¡ ,slh gSa tks 3 o 5 nksuksa ls foHkkT; gSa vr% oqQy izkf;drk
16 10 2
= + −
50 50 50
26 − 2 24 12
= = =
50 50 25

26-3 izkf;drk fl¼kar dk iz;ksx (Uses of Probability Theory)

buesa rks dksbZ lansg ugha fd izkf;drk fl¼kar vkt Hkh tqvkfj;ksa] rk'kcktksa iklk isaQdus okyksa vkSj lVksfj;ksa dk
,d fo'oklik=k lkFkh gS ysfdu bldk iklk mi;ksx mu {ks=kksa esa Hkh vR;fèkd gS tgk¡ ?kVuk,¡ vfuf'pr gksrh
gSa vkSj Hkkoh vuqeku yxkuk vko';d le>k tkrk gSA ,dkby cksjsy osQ 'kCnksa esa ¶izkf;drk fl¼kar osQoy vius
tUenkrkvksa (rk'kcktksa o iklk isaQdus okyksa) osQ fy, gh #fp dk fo"k; ugha jgk cfYd vk£Fkd fØ;kvksa dks
laiUu djus okys mu lHkh O;fDr;ksa m|ksxifr;ksa rFkk lsukifr;ksa osQ fy, Hkh mruk gh mi;ksxh gS ftudh liQyrk
lgh fu.kZ; ij vkèkkfjr gksrh gSA¸
izkf;drk fl¼kar xf.kr dh vR;Ur jkspd 'kk[kkvksa esa ls ,d gS vkSj vusd foKkuksa dh vkèkkjf'kyk gSA ØkDlVsu
,oa dkmMsu (Croxton and Cowden) osQ vuqlkj] ¶laHkkouk roZQ dk iz;ksx vkèkqfud ;qx esa tqvk] chek]
lS¼kfUrd HkkSfrd] izk.kh'kkL=k] vFkZ'kkL=k rFkk ,sls gh vusd vU; {ks=kksa esa Hkh fd;k tkrk gSA¸ la{ksi esa
(i) izkÑfrd ,oa lkekftd fo"k;ksa osQ y{k.kksa dh tkudkjh djus (ii) O;kolkf;d iwokZuqeku yxkus (iii) izfrp;u
iz.kkyh dks ifjiDork iznku djus rFkk (iv) chek lacaèkh dk;ks± dks fof'k"Vrk iznku djus esa izkf;drk fl¼kar dk
mi;ksx gksrk gSA
izkf;drk fl¼kar dk iz;ksx cuksZyh izes; rFkk cst izes; esa Hkh fd;k x;k gSμ

cuksZyh izes; ;k f}in izes; (Bernoulli’s or Binomial Theorem)
izkf;drk osQ lUnHkZ esa fdlh ;kn`fPNd iz;ksx dks ,d ckj djuk mldk ijh{k.k (Trail) dgk tkrk gSa ftl iz;ksx
osQ lEHko ifj.kke (Possible Outcome) osQoy nks gksrs gSa (;k mUgsa nks cuk fy;k tkrk gS)) ml iz;ksx dks ,d
ckj djus dks cuksZyh ijh{k.k (Bernoulli’s Trail) dgk tkrk gSA ;gka ,d ifj.kke dks liQyrk (Success) rFkk
nwljs ifj.kke dks vliQyrk (Failure) dgk tkrk gSA ,d iz;kl esa liQyrk dh izkf;drk dks p rFkk vliQyrk
dh izkf;drk dks q ls fu:fir fd;k tkrk gS tgka q + p = 1.
;fn p dk eku leku jgs vkSj cuksZyh iz;ksx dks n ckj nksgjk;k tk; ¹vFkkZr~ n ijh{k.k LorU=k (independent
trails) gksaº rks bl fej ijh{k.k dks LorU=k iqujko`Ùk cuksZyh ijh{k.k (Independent Repeated Bernoulli
Trails) dgk tkrk gSA bl ijh{k.k dh fo'ks"krk,a vxzfyf[kr gSaμ

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