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, the hne tht person holding tiket will win prize is winning is, nd his hne of not If p is the proility of the hppening of n event, the proility of its not hppening is p Insted of sying tht the hne of the hppening of n event is, it is sometimes stted tht the odds re '' to '' in fvour of the event, or '' to '' ginst the event If is the totl no of ses, eh eing eqully likely to our, nd of these '' re fvourle to the event, then the proility tht the event will hppen is, nd the proility tht it will not hppen is ex Wht is the hne of throwing numer greter thn 4 with n ordinry die whose fes re numered from to There re possile wys in whih the die n fll, nd of these two re fvourle to the event required required hne ex From g ontining 4 white nd lk lls mn drws t rndom, wht re the odds ginst these eing ll lk? The totl no of wys in whih lls n e drwn is 9 C nd the no of wys of drwing lk lls is C ; therefore the hne of drwing lk lls is C 9 C 4 9 8 4 Thus the odds ginst the event re to ex Find the hne of throwing t lest one e in simple throw with two die The possile no of ses is or An e on one die my e ssoited with ny of the numers on the other die, nd the remining numers on the first die my e ssoited with the e on the seond die, thus the numer of fvourle ses is Required hne ex4 Find the hne of throwing more thn in one throw with die A throw mounting to 8 must e mde up of,, nd this n our in wy, n e mde up of,, whih n our in wys, my e mde up of,, 4 nd,, eh of whih rrngements n our in wys Therefore the noof fvourle ses is = + + + = 0 nd the totl numer of ses is required hne 0 08
ex Wht is the proility tht digit seleted t rndom from the logrithmi tle is (i), (ii) or Vrious digits in the log tle re 0,,,, 4,,,, 8, 9, ie totl of 0 digits is used (i) The numer of fvourle ses for getting, out of 0 ll eqully likely ses is one Pro of getting is 0 (ii) The no of fvourle ses for getting or is Required Pro 0 ex Find out the proility of forming of 9 with the digits,,, 4,,,, 8, 9 when only numers of three digits re formed nd when ex (i) repetitions re not llowed (ii) repetitions re llowed (i) when repetitions re not llowed: The totl wys of forming numers of three digits with 9 given digits is 9 P 9 8 04 Of these, fvourle re two, ie nd 9 So the required proility is 04 (ii) When repetitions re llowed: Out of given 9 digits, digit numers n e formed in 9 9 9 9 wys Of these two re fvourle ses, Proility 9 In re where horses re running, the hne tht horse A will win is, tht B will win is 0 nd tht C will win is Assuming tht ded 8 het is impossile, find the hne tht one of them will win Proility tht A wins ( p ), tht ex8 ex9 B wins ( p ) nd tht C wins 0 ( p ) As ded het is impossile, these re mutully exlusive 8 events, so the hne tht one of them will win the re is p p p ie 4 0 8 0 Two lls re to e drwn from g ontining red nd white lls; find the hne tht they will oth e white Here ny one pir of lls is s likely to e drwn s ny other pir The totl numer of pirs is C, nd the numer of pirs whih re oth white is C, The required hne is therefore Find the hne of drwing white lls in suession f rom g ontining red nd white lls, the lls drwn not eing repled The hne of drwing white ll the first time is ; nd, hving drwn white ll the first time, there will e red nd white lls left, nd
ex 0 therefore the hne of drwing white ll the seond time will e Hene the hne of drwing two white lls in suession will e There re two gs, one of whih ontins red nd white lls nd the other red nd white lls, nd ll is to e drwn from one or other of the two gs; find the hne of drwing red ll The hne of hoosing the first g is, nd if the first g e hosen the hne of drwing red ll from it is ; hene the hne of drwing red ll from the first g is Similrly the 4 hne of drwing red ll from the seond g is 0 Hene s these events re mutully exlusive, the hne required is 4 0 0 ex In two gs there re to e put ltogether red nd 0 white lls, neither g eing empty How must the lls e divided so s to give person who drws one ll from either g (i) the lest hne nd (ii) the gretest hne of drwing red ll (i) The lest hne is when one g ontins only one white ll, nd the gretest hne is when one g EXERCISE ontins only one red ll, the hne eing nd respetively From pk of rds, two re drwn t rndom Find the hne tht one is knve nd the other queen ) ) 8 ) 8 d) A g ontins white, lk, nd 4 red lls Find the hne tht three lls drwn t rndom re ll white ) ) ) d) 4 of these Wht is the hne of throwing n e in only the first of two suessive throws with n ordinry die ) ) 4 ) d) of these 4 Three rds re drwn t rndom from n ordinry pk; find the hne tht they will onsist of knve, queen nd king ) ) of these ) d) If 8 oins re tossed, wht is the hne tht one nd only one will turn up hed? ) ) ) d) of these
In ertin town, the rtio of mles to femles is 000 : 98 If this tendeny is expeted to ontinue, wht is the hne tht newly orn y is mle? ) 000 98 ) 000 98 ) 98 000 d) 98 000 Wht is the hne tht lep yer, seleted t rndom, will ontin sundys? ) ) ) d) 4 8 Out of ll the integers from to 00, numer is seleted t rndom wht is the proility tht the seleted numer is not divisile y? ) 40 0 ) 4 0 ) 4 0 d) 4 0 of these 9 00 students ppered for two exmintions 0 pssed the first, 0 pssed the seond nd 0 pssed oth Find the proility tht student seleted t rndom hs filed in oth the exmintions ) ) ) d) 4 0 A hild is sked to pik up two lloons from ox ontining 0 lue nd red lloons Wht is the proility of the hild piking t rndom two lloons of different olours? ) ) ) 4 d) Answers: () () () 4 (d) () () () 8 (d) 9 () 0 () INEQUALITIES AND MAXIMA & MINIMA Any quntity is sid to e greter thn nother quntity when - is positive Thus, is greter thn - s - (-) = is +ve Also, is sid to e less thn when - is -ve Thus - less thn -, euse - - (-)= -, whih is -ve Zero must e regrded s greter thn ny -ve quntity nd less thn ny +ve quntity RULE - An inequlity will still hold fter eh side hs een inresed, diminished, multiplied or divided y the sme +ve quntity ie, if >, + > + - > - > is positive RULE - In n inequlity ny term my e trnsposed from one side to the other if its sign is hnged ie if - >, then > +, or ->- RULE - If the sides of ll the terms of n inequlity e hnged, the sign of the inequlity must e reversed ie if >, then < RULE 4 - If the signs of ll the terms of n inequlity e hnged, the sign of the inequlity must e reversed ie if >, then -< - or - < -, where is +ve RULE - If the sides of n inequlity e mutiplied y the sme -ve quntity, the sign of the inequlity must e rev ersed ie if >, then - < -; where is +ve n n RULE - If >, then, nd
RULE - n n n or +ve quntity n ; if n is The squre of every rel quntity is +ve nd therefore must e greter thn zero ie ( ) 0 ; z; Similrly, x y xy, x 0, y 0 Hene the rithmeti men of two +ve quntities is greter thn their geometri men RULE 8 - If the sum of two +ve quntities is given, their produt is gretest when they re equl; nd if the produt of two +ve quntities is given, their sum is lest when they re equl RULE 9 - If,,, k re n unequl quntities, then, k n n d k k ie ( k) n Therefore, the rithmeti men of ny numer of +ve quntities is greter thn their geometri men RULE 0- If nd re positive nd unequl, m m m, exept when 'm' is position proper frtion If m is positive integer or ny m m m negtive quntity If m is positiv e nd less thn,, m m m If there re n positive quntities,,, k, then k k n n m m m m m unless m is positive proper frtion RULE - RULE - It,, re +ve nd not ll equl, then ( + + ) (+ + ) > 9 nd, ( + ) ( + ) ( + ) > 8 x x OR ording s < OR > is x e positive or ording s > OR < if x is negtive e RULE - is less thn the d f gretest nd greter thn the lest, of the frtions, e d, f, RULE 4- If >x, >y, >z then + + + > x + y + z + nd > x y z RULE - RULE - n n n RULE - For ny positiv e integer n n ( ) n RULE 8 - RULE 9 - d d 4 4 4 4 4 RULE 0 - d 4d ex W hih of the two numer ( 00000 ) 000000 nd is greter?
000000 000000 ( 00000) ( ) 000000 Whih is greter thn (RULE ) ex whih of the two numers 000 000 nd 00 999 is greter? 00 000 999 000 00 000 000 000 000 00 whih is 00 000 00 000 999 00 ex Find tht min vlue of x 4x f or rel v lues of x ex 4 x 4x ( x ) A perfet squre is lwys positive, ie, it nnot e less thn zero the given expression is lest when ( x ) 0 min: vlue = Find the mximum v lue of 4x 4x for rel vlues of x 4x 4x 4 ( 4x 4x ) 4 ( x) the given expression is mximum when ( x ) is lest ie, when x 0 mximum vlue = 4 nd this ours when x 0 ex Solve x 4 0 8 4 x 4 x x ex Solve (-x + ) x x ex Solve ( x ) ( x 4) 0 ( x )( x ( 4) 0; x does not lie etween - 4 nd ex 8 Solve x 8x 0 ( x ) ( x ) 0; [ x ( )] [ x ( )] 0 x lies etween - nd - ex 9 Slove x 8x 0 ex 0 ( x )( x ) 0 ; x does not lie etween nd If w stisfies oth the following in equlities, nd w is n integer, wht vlues n w hve? (i) ( w 0) 4w 0 (ii) 8 w ( w ) w 0 4w 0; w 0 ex 8 w w ; w w lies etween -0 nd - x P ; Q x 8 x R 0 4x 4 ; S 0,,,, 4,
If x is n integer, list the memers of the following sets; () P S () Q S () R S (d) P Q S (e) ( P R) S P : x x ie P, 4, Q : x 8 xi e x 4; x 4, Q [,,, 0,,] R 0 4x 4 i e 4x 4, x, R [ 0,,,, ] Therefore, P S [, 4, ] Q S [ 0,,, ] R S [ 0,,,, 4, ] P Q S [ ] ( P R) S [ 0,,,, 4, ] (), 4, () 0,,, () 0,,,,4, (d) (e) 0,,,, 4, ex Between wht vlues of x, is the expression 9 x x positive? Let y denote the given expression, y ( x 9x ) ( x ) ( x ) ( x ) ( x) ( x ) ( x) when x ; x is -v e nd ( - x) is +ve, y is negtive When x ut ; x +ve nd x is ve y is positive when x, x is +ve nd -x is -ve; y is -ve The given expression is positive only s long s x is etween nd ex Find the gretest v lue of ( x) ( x) 4 for ny rel vlue of x The given expresssion is gretest when: x x OR x 4 Thus the gretest v lue is 8 4 m n [Note : p will e gretest when the f tors m n p ] Rememer : In order tht x x my e lwys +ve, 4 must e - ve or 0, nd must e +ve In order tht x x my e lwys negtive, 4 must e -ve or 0 nd must e -ve ex 4 Find the gretest v lue of x for rel vlues of x x x x Let Y ; then x x
yx ( y ) x y 0 If x is rel, ( y ) 8y ( y ) must e positive (+y) ( y) must e +ve Hene y must lie etween nd nd its gretest vlue is ex : Find the lest vlue of x 4y x y if, x nd y re positive 4 Let A x ; B y; ex 4 A x y B 9 4 A B Now, x + 4y = A + B lso, AABBB = The lest vlue of A+A+B+B+B will e so when ll the quntities re equl (RULE 8) A B ; Lest vlue of x 4y 0 If x my hve ny rel vlue, find whih is greter, x x OR x 0 x x x 0 hs ftor x- x x x 0 ( x )( x x 0) ( x )[( x ) ] The Seond f tor is lwys positive, hene x if x EXERCISE x is grete If x e positive, find the gretest vlues of ( x)( x ) ) ) ) 8 d) 0 of these If x e rel, find the mx nd min vlues x of x x x ), ), ) 4, 4 d) 4, d), of these Find the mx vlue of rel vlues of x ) ) x x ) 4 d) 4 Find the min vlue of x x ) ) ) 4 d) for Find the gretest vlue of x y when x y ) 49 4 ) ) 4 d) 8 Answers: 4 4 () () () 4 () (d)