Unique Solutions R. 4. Probability. C h a p t e r. G l a n c e

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Virginia Welch
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1 . Proility C h p t e r t G l n e Rnom Experiment : An t in whih ll possile (outomes) results re known in vne ut none of them n e preite with ertinty is lle rnom experiment. For e.g. When we toss oin, we know the results of this t. Either he or til omes up, ut we n't sy for sure whether he omes up or til. Outomes : The results of the rnom experiment re lle outomes. Eqully likely outomes : Outomes re si to e eqully likely if we hve no reson to elieve tht one is more likely to our thn other. Smple spe : Set of ll possile outomes of rnom experiment is lle the smple spe enote s 'S'. Numer of elements in the smple spe is enote n(s). Event : Every suset of smple spe is lle n event. Types of Events : () Certin/Sure event : The event whih ontins ll the smple points of the smple spe is lle ertin event. () Impossile/Null event : An event tht oes not ontin ny smple point of smple spe S is lle n impossile event. () Elementry event : An event onsisting of only one smple point of smple spe is lle n elementry event. () Complement of n event : Let A e n event of smple spe S. Then the set of ll elements of S, whih re not in A is lle the omplement of A, enote s A or A. A A' = S. Also n(a) + n(a ) = n(s) (e) Mutully exlusive events : Let S e smple spe n A n B e two events of S. Then event A n B re lle mutully exlusive if A B = (f) Exhustive events : Two or more events re exhustive events, if their union is the smple spe. Note : (i) If A B =, A n B re mutully exlusive events. (ii) If A B = S, A n B re exhustive events. (iii) If A B = n A B = S, then A n B re omplementry events. (iv) A A =, A A' = S, A n A' re exlusive n exhustive events. Proility of n events.

2 ALGEBRA - S.S.C. 7 The proility of n event A of smple spe S, is P(A) = 0 P(A) If S is the finite smple spe n A is n event of S, the P(A) + P(A ) =, where A is the omplementry event of A. MULTIPLE CHOICE QUESTIONS (MCQ's) (Eh question rries one mrk). When oins re tosse simultneously, the numer of elements in the smple spe is Numer of fe rs in pk of rs is oins re tosse simultneously; A is the event of getting no he; then P(A) is..... Two ie re thrown simultneously. E is the event tht sum of numers on the uppermost fe is t lest 0; then n(e) is A g ontins re, white n green lls. One ll is rwn t rnom. E is the event tht the ll rwn is re; then n(e) is A oin is tosse n ie is thrown simultneously. A is n event of getting he n n even numer; then n(a) is S is finite smple spe. A is n event of the smple spe n A is its omplementry event; then n(a) n(a ) = n(s) n(a) + n(a ) = n(s) n(a) + n(a ) = n(a) + n(a ) = 0. Two oins re tosse; then the proility tht t lest one he turns up is A ie is thrown, the proility of getting perfet squre is Proility

3 Unique MCQ's 0. Two ie re rolle simultneously. A is n event tht sum of the numers is ivisile y. Then P(A) is..... A ox ontins 0 rs, numere from to 0. One r is rwn t rnom. B is the event tht the r rwn ers numer whih is perfet squre, then n(b) is, igit numers re forme from the igits 0,,,, when igits re not repete. B is the event tht the numer forme is greter thn 0, then n(b) is A r is rwn t rnom from pk of rs. rs the proility of getting lk r is..... A r is rwn t rnom from well-shuffle pk of rs. The proility tht the r rwn is imon is oins re tosse. A is the event of getting t the most one he then A =? {HH, HT, TH, TT} {HH, HT, TH} {HT, TH, TT} {HT, TH} 6. An unise ie is thrown. A is the event tht prime numer omes up, then A =? {,,, 5} {,, 5} {,, 5} {,, } 7. Two ie re rolle simultneously. A is n event tht prout of numers on the uppermost fe is, then P(A) =?. S is finite smple spe. A is n event of the smple spe n A is its omplementry event then.... (A ) = A A A = S n(a) + n(a ) = 0 n(a) + n(a ) =. One r is rwn from well-shuffle pk of rs. The proility of getting spe is S = {,,,, 5, 6, 7,,, 0} A = {,, 6,, 0} A =... {,, 5, 7, } {,, 5} {,, 5} {5, 7, } 6 Proility

4 ALGEBRA - S.S.C.. A oin is tosse n ie is thrown simultneously. The numer of smple point n(s) is A r is rwn from pk of plying rs. Wht is the proility of getting n e? 6. A g ontins re lls, lue lls n 5 green lls. Wht is the proility tht ll pike up t rnom is not lue ll?. () S = {HHH, HHT, HTH, HTT, THT, TTH, THH, TTT} n(s) =. () Numer of fe rs = Kings + Queens + Jks =. () S = {HHH, HHT, HTH, HTT, THT, TTH, THH, TTT} n(s) = n(a) = A = [TTT] n P(A) = (A) n(s) = Answers. A ie is thrown. If A is the event of getting sore on the upper surfe whih is ivisile y 5, then A is..... () E = {(, 6), (6, ), (5, 5), (5, 6), (6, 5), (6, 6)} n(s) = 6 5. () E = {R, R, R } n(e) = 6. () A = {H, H, H 6 } n(a) = 7. () A A = S n(a) + n(a ) = n(s) ertin event n impossile event n elementry event mutully exlusive event. 5. Whih of the following numers nnot e the proility of n event? 0 0. () When ie re rolle, n(s) = 6 Let A e the event tht sum of the numers is ivisile y. A = {(, 6), (6, ), (, 5), (5, )} n(a) = P(A) = 6 = Proility 0 7. () When oins re tosse. S = {HH, HT, TH, TT} n(s) = Let E e the event tht tlest one he turns up. E = {HH, HT, TH} n(e) = P(E) =. () When ie is rolle. S = {,,,, 5, 6} n(s) = 6 A : Event of getting perfet squre. A = {, ) n(a) = P(A) = 6 =

5 0 Unique MCQ's. () A = {,,, 6) n(a) =. () B = {,, } n(b) =. () A r is rwn from pk n(s) = E : Event tht the r rwn is lk n(e) = 6 P(E) = 6. () A r is rwn from pk. n(s) = There re imons in pk. P(E) = 5. () At the most one he. So one he or no he. A = {HT, TH, TT} 6. () A = {,, 5} 7. () A = {(, ), (, ), (, 6), (6, )} n(a) = But, n(s) = 6 P(A) = 6 =. () (A ) = A. () n(s) = ; No. of spe = P(A) = = 0. () A = {,, 5, 7, }. () S = {H, H, H, H, H 5, H 6 T, T, T, T, T 5, T 6 } n(s) =. () n(s) = No. of e rs = P(A) = =. () n(s) = n(s) = n(a) = Prolems For Prtie. When n unise ie is thrown, n(s) is.... P(A) = =. () S = {,,,, 5} A = { 5 } n(a) = Only one element elementry event 5. () Proility of ny event lies etween 0 n oth inlusive i.e. 0 P(A) 6. If S = {HH, HT, TH, TT}, then n(s) =..... A suset of smple spe is lle.... n event out ome proility rnom experiment. If the smple spe S = {,,,, 5, 6} n the event A = {,, 5} then A =.... {,, } {,, 6} {, } {, } 5. Proility of n impossile event is If A is n event of smple spe S, then P(A) =.... Proility

6 ALGEBRA - S.S.C. 7. Proility of unertin event is When oin is tosse, proility of getting he is An unise ie is thrown, then the proility of getting numer less thn is S is finite smple spe, A is n event n A is its omplementry event, then.... P(A) P(A ) = P(A) + P(A ) = P(A) + P(A ) = 0 P(A) + P(A ) =. A ox ontins 0 rs numere from to 0. One r is rwn t rnom. E is the event tht the r rwn is multiple of 5, then n(e) = A r is rwn t rnom from well shuffle pk of rs. The proility of getting fe r is Answers Proility

Probability. b a b. a b 32.

Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,