OCR Maths S1. Topic Questions from Papers. Probability. Answers

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1 OCR Maths S Topic Questions from Papers Probability Answers PhysicsAndMathsTutor.com

2 = 0 A 0, cao (i) B(, /8) B(, /8) stated or / 8, / 8 seen and sum of powers = C (/8) (/8) = /8 or 0. A Correct expression Answer, a.r.t. 0. ISW (ii) ½ p = / 8 or / 8 / / or / 8 x p = ¾ AG A ¾ correctly obtained. Must see explicit step. (iii) ½ p = / p = / A PhysicsAndMathsTutor.com Verification eg / x / = / 8 or / 8 / / = / : A or / / / or / x Answer / or a.r.t. 0.7 (Q, Jan 00) (i) In qus & prod means product of two probabilities / 7 or 0.7 ( sfs) (ii) / 8 x / 7 + / 8 x / 8 = / 8 or 0.9 ( sfs) A : one correct prod or add any two prods : all correct (iii) / 8 x / 8 + / 8 x / 7 = / 8 or 0.0 ( sfs) A Total 7 : one correct prod or add any two prods : all correct (i) 7! : 7!/(a factorial); or (Q, (! x June (!)) 00) 7i (i) ii iii / 9 or 7 / 9 oe seen / 9 or / 9 oe seen / 8 or 7 / 8 oe seen Correct structure All correct / 0 x 7 / / 0 x / / 0 x / 9 / or 0.9 oe / 0 x / 9 x 7 / / 0 x / 9 M A M ie 8 correct branches only, ignore probs & values including probs and values, but headings not req d or / 0 x 7 / / 0 or / 0 x / 9 : one correct prod or any prod + 7 / 0 or / 0 x / 9 : one correct prod / 0 or 0. oe A cao No ft from diag except: with replacement: (i) structure: (ii) 9 / 00 : B (iii) 0.: B Total 8i Med = cao (Q7, Jan 007)

3 Total ia (ia) / oe B W MR: max (a)b0(b)(c) b c P(BB) + P(WB) attempted = / 0 / 9 + / 0 / 9 or / + / = / oe Denoms 9 & 8 seen or implied / 9 / 8 + / 9 / 8 A Or / 0 / 9 OR / 0 / 9 correct NB / 0 / 0 + / 0 / 0 = / : M0A0 Or / as numerator Or / / 0 Or / 0 / 9 / 8 + / 0 / 9 / 8 above + / 0 / 9 / 8 + / 0 / 9 / 8 ii = / oe A P(Blue) not constant or discs not indep, so no Total 8 May not see wking Prob changes as discs removed Limit to no. of discs. Fixed no. of discs Discs will run out Context essential: disc or blue NOT fixed no. of trials NOT because without repl Ignore extra (Q, June 007) 8i (i) \/0.0 (= 0.) ( their \ /0.0) = 0. ii p seen for either p( p) = 0. or p( p) = 0. oe p p + 0.(= 0) or p - p + 0.(= 0) ± \ /((-) 0.) or ± \ /((-) 0.) or (p 0.7)(p 0.)=0 or (0p 7)(0p )=0 p = 0.7 or 0. A A pq= 0. or pq =0. Allow pq=0. or opp signs, correct terms any order (= 0) oe Correct Dep Any corr subst n or fact n Omit in nd line: max M0M0A0 One corr ans with no or inadeq wking: SC eg = 0. p = 0.7 or 0. p + pq + q = p + q = 0.8 } p = 0./q } p - 0.8p = 0 corr subst n or fact n Total 8 9ia / / p seen p( p) = 0. or p( p) = 0. p p = -0. p p + 0. = oe } OR (p 0.) 0. = -0. oe } (p 0.) = 0.0 (p 0.) = ±0.0 p = 0. or 0.7 A (Q8, June 007)

4 Total i (i) 0. x p = 0. or 0. / 0. or / 0 oe p = 0. oe A ii 0. x ( their 0.) oe eg 0 / 00 8 / 0 PhysicsAndMathsTutor.com or or 0.8 or 8 seen Not unless ans to (i) is or 8% oe Aft Total ia Binomial stated or implied by use of tables or 0. a x 0.8 b, a+b (Q, = Jan 008) 87 (i) (i) (ii) (iii) / + ( / ) M = / A / routes clearly implied out of 8 possible (equiprobable) routes / A Total 7 or ( / ) or / + ( / ) or / + ( / ) or / + ( / ) or / / M or / / or / / or / / or / - / but / / M0 ( ) or or oe M or P(&twice) P(twice) stated or prob Whatever st, only one possibility on nd M /, no wking A /, no wking M0 (Q8, Jan 009)

5 8ia 8 (ia) 8 / 9 or / 9 seen 7 / 8 or / 8 seen structure correct ie branches all correct incl. probs and W & R regardless of probs & labels (or branches with correct 0s & s) b / / 0 / / 0 8 / 9 / 8 = / 0 M A any correct terms added 9 / 0 8 / 9 7 / 8 9 / 0 8 / 9 7 / 8 iia 9 / 0 8 / 9 = 9 / 0 oe A 9 / 0 8 / 9 / / 0 8 / 9 7 / 8 or / / 0 b (P(X = ) = / 0 ) 9 / 0 / 9 = / 0 Σxp = 7 / 0 or.8 A A or ( / / 0 ) or probs of / 0 A > terms, ft their p s if Σp = NB: 9 / 0 =.8 no mks ia ib iia b Total With replacement: Original scheme / / 0 / 0 + ( 9 / 0 ) / 0 or ( 9 / 0 ) ( 9 / 0 ) or ( 9 / 0 ) / 0 + ( 9 / 0 ) 9 / 0 Original scheme But NB ans.8( ) A0A0 (Q8, June 009) Total [7] 7 9 (i) (i) / 7 B num, denom /7xp (ii) seen or implied 9 9 seen or implied M num, denom = 8 7 or 0. ( sfs) oe Allow for 9/9x or + wrong p A Total [] (Q7, Jan 00)

6 0 (i) i = 0.88 ii p + ( p) p = 0. or p p = 0. A or 0. + prod of probs Condone or or p + p ( p) + ( p) p prod of P s or Condone p + p p, but p + qp = 0. M0 p p + 0. = 0 0. (p 0.)(p.7) = 0 or p = oe A Correct QE = 0 Condone omission of = 0 Correct method for their -term QE or ( p) = 0.9 A p = ± 0.7 must have ± p = 0. A Total 7 Not p = 0. or.7 Correct ans from correct but reduced wking or T & I or verification or no wking: mks Ans p = 0. or.7 from correct but reduced wking or T & I or no wking: A0 Ans p = 0. following correct wking except other solution incorrect: BOD mks 0. (eg p = so p = 0. or -. so p = 0.: mks)) p = 0. from wrong wking but correct verification: BOD mks p = 0. from wrong wking alone: M0A0M0A0 (Q, Jan 0) Total (i) If done with replacement, no marks in any part of this question. i All correct probs correctly placed, for correct probs anywhere matching labels, if any B ii : two of these products (or their results) added Allow B with missing labels but only if probs consistently placed, ie R above B throughout (Q, June 0) Total 8 Incorrect p (eg cubical die means 8 sides hence p = 8 ): can gain all B & M marks. (i) 8i / oe or 0. ( sfs) ii ( / ) 7 / alone M for ( / ) 8 / alone iii iv = 0.0 ( sfs) or ( / ) 8 oe alone = 0. ( sfs) or A A NB If more than products are added (eg P( < X < ): no marks P(X < 8), with exactly 8 correct terms NOT ( ) 8, NOT ( ) 8. ( / ) 9 / + ( / ) 0 / + ( / ) / + ( / ) / (= ) M M for all correct or M for of these added or these plus extra or or or 0.9 or 0. ( / ) 9 ( / ) or ( / ) [ ( / ) 9 ] M or ( / ) 8, 9 or 0 (, or / ) or ( / ), or [( ( / ) 8, 9 or 0 ] M = 0.00 ( sfs) A Total 0 Total 7 marks or for > of these terms or values seen; ignore incorrect Allow 0. with wking or ±[( / ) 9 ( ( / ) )] or ± [-( / ) 9 ( / ) ] (Q8, June 0)

7 (i) Top: branches, & R, B shown Bottom: st branch: prob = or, & R shown (ii) no nd branch OR branch with prob = 0 or 0 or ( ) or seen + ( ) = oe [] A [] consistent allow eg ignore any rd layer branches all correct cao Can see important data items/measures Any missing label(s) on first three branches, subtr once No label needed on zero branch, if drawn. or or M ft incorrect tree dep probs < if rd tree prob =, (ii)a0 if rd tree prob, (ii)m0a0 (iii) + ( ) or or 0. all correct or ( ) all correct NB!! = M0A0 but ( + ) M0 = or 0.8 oe A [] May be seen without working A cao ft incorrect tree: (iii) A0 (Q, June 0) ( 0.) (= 0.8) 0.8 or 0.8 or 7 0. (or result of these)(poss 00) ( 0.8 only scores if using, not score of. Similarly for 0.8). can be implied, eg by or 0 0.(or result of these)(poss 00) or, using exp no. of s & s 8 or AND (poss 00) (or. AND.0 or AND 0) AND (poss 00) (or AND.9 or 00 AND 90) 00 AND (NB is insuff)..0 or 0 must be attempt gain on,,, loss on, E(profit for 00 rolls) = ( )0 dep any A [].9 or must be attempt receipt payout on, E(profit for 00 rolls) = ( )0 NB 00 ( ) = 00 8 = M0M0A0 Eg: ( )=0 M0A0 Mark one method only Must be matched pair eg 00 0 or 90: M0M0A0 7 (i) (a) (Q, June 0) (i) = 7 or 0.87 ( sf) (ii) 0.p or equiv seen p = 0.8 p = 0. [] A [] A [] (= 8 or 0.7) 8 Tree diag alone insufficient for mark. Or 0.p = alone insufficient NB = 0.8 seen at the end is probably a check, not an answer. But if 0. seen and 0.8 is very clearly indicated as the ans then A0 (Q, Jan 0)