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PZz`©k Aa¨vq

m¤¢vebv

cvV m¤úwK©Z MyiæZ¡c~Y© welqvw`

¾ m¤¢vebvi mv‡_ RwoZ wKQz k‡ãi aviYv :

¾ ˆ`e cixÿv (Random Experiment) : hLb †Kv‡bv cixÿvi m¤¢ve¨ mKj djvdj Av‡M †_‡K Rvbv _v‡K wKš‘ cixÿvwU‡Z †Kv‡bv GKwU wbw`©ó †Póvq Kx djvdj Avm‡e Zv wbwðZ K‡i ejv hvq bv, G‡K ˆ`e cixÿv e‡j| †hgb, GKUv gy`ªv wb‡ÿc cixÿvi m¤¢ve¨ djvdj Head (H), Tail (T) n‡e, Zv Avgiv Av‡M †_‡KB Rvwb wKš‘ †Kvb djvdjwU NU‡e gy`ªvwU wb‡ÿ‡ci c~‡e© Zv Avgiv wbwðZ K‡i ej‡Z cvwi bv| myZivs gy`ªv wb‡ÿc cixÿv GKUv ˆ`e cixÿv|

¾ NUbv (Event) : †Kv‡bv cixÿvi djvdj ev djvd‡ji mgv‡ek‡K NUbv e‡j| D`vniY¯^iƒc GKUv Q°v wb‡ÿc cixÿvq Ô3' cvIqv GKUv NUbv| Avevi †Rvo msL¨v cvIqvI GKwU NUbv|

¾ mgm¤¢ve¨ NUbvewj (Equally Likely Events) : †Kv‡bv cixÿvi NUbv¸‡jv NUvi m¤¢vebv mgvb nq A_©vr GKwU AciwUi †P‡q †ewk ev Kg m¤¢ve¨ bv nq, Z‡e NUbv¸‡jv‡K mgm¤¢ve¨ e‡j| †hgb GKUv wbi‡cÿ gy`ªv wb‡ÿ‡c †nW ev †Uj Avmvi m¤¢vebv mgvb mgvb| myZivs †nW Avmv I †Uj Avmv NUbv `yBwU mgm¤¢ve¨ NUbv|

¾ ci¯úi wew”Qbœ NUbvewj (Mutually Exclusive Events) : †Kv‡bv cixÿvq hw` GKUv NUbv NU‡j Ab¨Uv A_ev Ab¨ NUbv¸‡jv bv NU‡Z cv‡i Z‡e D³ NUbv¸‡jv‡K ci¯úi wew”Qbœ NUbv e‡j| †hgb, GKUv wbi‡cÿ gy`ªv wb‡ÿc Ki‡j †nW Avmv ev †Uj Avmv `yBwU wew”Qbœ NUbv| †Kbbv †nW Avm‡j †Uj Avm‡Z cv‡i bv| Avevi †Uj Avm‡j †nW Avm‡Z cv‡i bv| A_©vr †nW I †Uj GKmv‡_ Avm‡Z cv‡i bv|

¾ AbyK‚j djvdj (Favourable Outcomes) : †Kv‡bv cixÿvq GKUv NUbvi ¯^c‡ÿi djvdj‡K D³ NUbvi AbyK‚j djvdj e‡j| †hgb, GKwU Q°v wb‡ÿc Ki‡j we‡Rvo msL¨v nIqvi AbyK‚j djvdj 3 wU|

¾ bgybv‡ÿÎ (Sample Space) : †Kv‡bv ˆ`e cixÿvi m¤¢ve¨ mKj djvdj wb‡q MwVZ †mU‡K bgybv‡ÿÎ e‡j| GKUv gy`ªv wb‡ÿc Ki‡j `yBwU m¤¢ve¨ djvdj cvIqv hvq| h_v †nW (H) I †Uj (T). GLb S Øviv G cixÿ‡Yi djvd‡ji †mU‡K m~wPZ Ki‡j bgybv‡ÿÎ S = {H, T}.

¾ bgybv we›`y (Sample Point) : bgybv‡ÿ‡Îi cÖwZwU Dcv`vb‡K djvd‡ji bgybv we›`y e‡j| GKUv gy`ªv GKevi wb‡ÿc cixÿvq bgybv‡ÿÎ S = {H, T} Ges GLv‡b H, T cÖ‡Z¨‡KB GK GKwU bgybv we›`y|

¾ `ywU we‡kl ai‡bi NUbv:

wbwðZ NUbv : †Kv‡bv cixÿvq †h NUbv Aek¨B NU‡e G‡K wbwðZ NUbv e‡j|

D`vniY : AvMvgxKvj m~h© c~e© w`K †_‡K DV‡e AvR m~h© cwðg w`‡K A¯Í hv‡e| G¸‡jvi cÖ‡Z¨KwU wbwðZ NUbvi m¤¢vebv|

¾ Am¤¢e NUbv : †Kv‡bv cixÿvq †h NUbv KL‡bv NU‡e bv A_©vr NU‡Z cv‡i bv G‡K Am¤¢e NUbv e‡j| Am¤¢e NUbvi m¤¢vebv me mgq k~b¨ nq|

D`vniY : AvMvgxKvj m~h© cwðg w`‡K DV‡e A_ev m~h© c~e©w`‡K A¯Í hv‡e, Am¤¢e NUbvi m¤¢vebv k~b¨| G¸‡jvi cÖ‡Z¨KwU NUbvB Am¤¢e NUbv|

GK bR‡i G Abykxjbx‡Z e¨eüZ m~Îmg~n :

l †Kv‡bv NUbv NUvi m¤¢vebv =

l †Kv‡bv NUbv NUv Ges bv NUvi m¤¢vebvi †hvMdj|

l †h‡Kv‡bv NUbvi m¤¢vebv 0 †_‡K 1 Gi g‡a¨ _vK‡e|

l A NUbv NUvi m¤¢vebv P(A) n‡j O < P(A) < 1

l hw` †Kv‡bv GKwU cixÿ‡Yi †gvU m¤¢ve¨ djvdj A Ges Aci GKwU cixÿ‡Yi †gvU m¤¢ve¨ djvdj B nq Zvn‡j cixÿv `yBwU GK‡Î msNwUZ n‡j †gvU m¤¢ve¨ djvdj = A ´ B

l bgybv‡ÿ‡Îi AvKvi †ek eo n‡j Probability tree Gi mvnv‡h¨ bgybv‡ÿÎ ˆZwi Kiv hvq Ges wewfbœ NUbvi m¤¢vebv wbY©q Kiv hvq|

Abykxjbxi cÖkœ I mgvavb

1. GKwU Q°v gvi‡j 3 DVvi m¤¢vebv †KvbwU?

˜ L M N

wb‡Pi Z_¨ †_‡K (2Ñ3) b¤^i cÖ‡kœi DËi `vI :

GKwU _wj‡Z bxj ej 12wU, mv`v ej 16wU Ges Kv‡jv ej 20wU Av‡Q| ˆ`efv‡e GKUv ej †bIqv n‡jv|

2. ejwU bxj nIqvi m¤¢vebv KZ?

K L M ˜

3. ejwU mv`v bv nIqvi m¤¢vebv KZ?

K ˜ M N

wb‡gœi Z_¨ †_‡K (4Ñ6) b¤^i cÖ‡kœi DËi `vI :

GKwU gy`ªv‡K wZbevi wb‡ÿc Kiv n‡jv|

4. me©vwaK evi H Avmvi m¤¢vebv KZ?

˜ 1 evi L 2 evi M 3 evi N 4 evi

5. me‡P‡q Kg msL¨K evi T Avmvi m¤¢vebv KZ?

K 0 L M 1 N 2

[ mwVK DËi : ]

6. PÆMÖvg AvenvIqv Awd‡mi wi‡cvU© Abyhvqx 2012 mv‡ji RyjvB gv‡mi 1g mßv‡n e„wó n‡q‡Q †gvU 5 w`b| †mvgevi e„wó bv nIqvi m¤¢vebv KZ?

K ˜ M N 1

cÖkœ \ 7 \ 30wU wU‡K‡U 1 †_‡K 30 ch©šÍ µwgK b¤^i †`qv Av‡Q| wU‡KU¸‡jv fv‡jvfv‡e wgwk‡q GKwU wU‡KU ˆ`efv‡e †bqv n‡jv| wU‡KUwU (i) †Rvo msL¨v (ii) Pvi Øviv wefvR¨ (iii) 8 Gi †P‡q †QvU (iv) 22 Gi †P‡q eo-nIqvi m¤¢vebv¸‡jv wbY©q Ki|

mgvavb : wU‡KU¸‡jv‡Z 1 †_‡K 30 ch©šÍ µwgK b¤^i †`Iqv Av‡Q| myZivs mgMÖ m¤¢ve¨ djvdj 30|

i. 1 †_‡K 30 ch©šÍ µwgK b¤^‡ii g‡a¨ †gvU †Rvo msL¨v 15wU| †Rvo msL¨v cvIqvi AbyK‚j NUbv = 15wU

\ P †Rvo msL¨v =

= = (Ans.)

ii. 1 †_‡K 30 ch©šÍ µwgK b¤^‡ii g‡a¨ 4 Øviv wefvR¨ msL¨v 7wU h_v : 4, 8, 12, 16, 20, 24, 28| myZivs AbyK‚j djvdj 7wU|

\ P (4 Øviv wefvR¨) = (Ans.)

iii. 1 †_‡K 30 ch©šÍ µwgK b¤^‡ii g‡a¨ 8 Gi †P‡q †QvU µwgK b¤^i 7wU| h_v : 1, 2, 3, 4, 5, 6, 7

myZivs AbyK‚j djvdj 7wU|

\ P (8 Gi †P‡q †QvU) = (Ans.)

iv. wU‡KU¸‡jv‡Z 22 Gi †P‡q eo µwgK b¤^i 8wU h_v : 23, 24, 25, 26, 27, 28, 29, 30.

myZivs AbyK‚j djvdj 8wU|

\ P (22 Gi †P‡q eo) = = (Ans.)

cÖkœ \ 8 \ †Kv‡bv GKwU jUvwi‡Z 570wU wU‡KU wewµ n‡q‡Q| iwng 15wU wU‡KU wK‡b‡Q| wU‡KU¸‡jv fv‡jvfv‡e wgwk‡q GKwU wU‡KU ˆ`efv‡e cÖ_g cyi¯‹v‡ii Rb¨ †Zvjv n‡jv| iwn‡gi cÖ_g cyi¯‹vi cvIqvi m¤¢vebv KZ?

mgvavb : †gvU wU‡KU wewµ n‡q‡Q = 570wU

iwng wU‡KU wK‡b‡Q = 15wU

\ iwn‡gi cÖ_g cyi¯‹vi cvIqvi m¤¢vebv = = (Ans.)

cÖkœ \ 9 \ GKUv Q°v GKevi wb‡ÿc Kiv n‡j †Rvo msL¨v A_ev wZb Øviv wefvR¨ msL¨v DVvi m¤¢vebv KZ?

mgvavb : GKUv Q°v wb‡ÿc cixÿvq bgybv †ÿÎ : {1, 2, 3, 4, 5, 6}

\ mgMÖ m¤¢ve¨ djvdj 6

GKUv Q°vq †Rvo msL¨v Av‡Q 3wU, h_v : {2, 4, 6}| myZivs †Rvo msL¨v cvIqvi AbyK‚j djvdj 3wU

\ P (†Rvo msL¨v) = = =

Avevi, 3 Øviv wefvR¨ msL¨v 2wU, h_v : {3, 6}| myZivs 3 Øviv wefvR¨ msL¨v cvIqvi AbyK‚j djvdj 2wU|

\ P (3 Øviv wefvR¨ msL¨v) = = =

†h‡nZz †Rvo msL¨v Ges 3 Øviv wefvR¨ msL¨v Df‡qi g‡a¨ 6 AšÍfy©³| myZivs †Rvo msL¨v Ges 3 Øviv wefvR¨ msL¨v cvIqvi AbyK‚j djvdj 1wU|

\ P (†Rvo msL¨v Ges 3 Øviv wefvR¨ msL¨v) =

\ P (†Rvo msL¨v A_ev 3 Øviv wefvR¨) =

P (†Rvo msL¨v) + P (3 Øviv wefvR¨) - P (†Rvo msL¨v Ges 3 Øviv wefvR¨)

= + - = = = (Ans.)

cÖkœ \ 10 \ †Kv‡bv GKwU ¯^v¯’¨ †K‡›`ªi wi‡cvU© Abyhvqx 155 wkï Kg IR‡bi, 386 wkï ¯^vfvweK IR‡bi Ges 98wU wkï †ewk IR‡bi Rb¥ †bq| GLvb n‡Z GKwU wkï ˆ`efv‡e wbe©vPb Ki‡j wbe©vwPZ wkïwU †ewk IR‡bi n‡e Gi m¤¢vebv KZ?

mgvavb : ¯^v¯’¨ †K‡›`ª †gvU wkï Rb¥ †bq (155 + 386 + 98) = 639 Rb| ˆ`efv‡e GKRb wkï wbe©vPb Ki‡j 639 R‡bi †h †Kv‡bv GKRb wkï Avm‡Z cv‡i|

\ mgMÖ m¤¢ve¨ djvdj = 639

†ewk IR‡bi wkïi AbyK‚‡j m¤¢ve¨ djvdj = 98

myZivs †ewk IR‡bi wkï Rb¥ †bIqvi m¤¢vebv = (Ans.)

cÖkœ \ 11 \ `yB nvRvi jvB‡mÝ cÖvß WªvBfvi GK eQ‡i wbgœwjwLZ msL¨K UªvwdK AvBb f½ K‡i|

UªvwdK AvBb f‡½i msL¨v

WªvBfv‡ii msL¨v

5 ev Zvi AwaK

1910

GKRb WªvBfvi‡K ˆ`efv‡e wbe©vPb Ki‡j WªvBfviwUi 1wU AvBb f½ Kivi m¤¢vebv KZ? WªvBfviwUi 4 Gi AwaK AvBb f½ Kivi m¤¢vebv KZ?

mgvavb : †gvU WªvBfv‡ii msL¨v

= (1910 + 46 + 18 + 12 + 9 + 5) Rb = 2000 Rb

1wU UªvwdK AvBb f½ K‡i‡Q Ggb WªvBfv‡ii msL¨v = 46

Avevi,

4 Gi AwaK AvBb f½ K‡i‡Q Ggb WªvBfv‡ii msL¨v = 5

cÖkœ \ 12 \ †Kv‡bv GKwU d¨v±wi‡Z wb‡qvMK…Z †jvK‡`i Kv‡Ri aiY Abyhvqx wbgœfv‡e †kªwYK…Z Kiv hvq :

†kªwYKiY

msL¨v

e¨e¯’vcbvq

cwi`k©K wn‡m‡e

Drcv`b Kv‡R

Awdwmqvj Kv‡R

157

1473

215

GKRb‡K ˆ`efv‡e wbe©vPb Ki‡j †jvKwU e¨e¯’vcbvq wb‡qvwRZ Gi m¤¢vebv KZ? †jvKwU e¨e¯’vcbvq A_ev Drcv`b Kv‡R wb‡qvwRZ Gi m¤¢vebv KZ?

†jvKwU Drcv`b Kv‡R wb‡qvwRZ bq Gi m¤¢vebv KZ?

mgvavb : d¨v±wi‡Z wb‡qvMK…Z †gvU †jv‡Ki msL¨v =

(157 + 52 + 1473 + 215) Rb = 1897 Rb

e¨e¯’vcbvq wb‡qvwRZ †jv‡Ki msL¨v 157 Rb

\ †jvKwUi e¨e¯’vcbvq wb‡qvwRZ nIqvi m¤¢vebv = (Ans.)

Avevi, Drcv`b Kv‡R wb‡qvwRZ †jv‡Ki msL¨v = 1473 Rb

\ Drcv`b Kv‡R wb‡qvwRZ nIqvi m¤¢vebv =

myZivs †jvKwUi e¨e¯’vcbvq A_ev Drcv`b Kv‡R wb‡qvwRZ nIqvi m¤¢vebv = + = = (Ans.)

Drcv`b Kv‡R wb‡qvwRZ bq Ggb †jv‡Ki msL¨v

= (157 + 52 + 215) = 424

\ †jvKwU Drcv`b Kv‡R wb‡qvwRZ bq Ggb m¤¢vebv = (Ans.)

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