Q Scheme Marks AOs. Notes. Ignore any extra columns with 0 probability. Otherwise 1 for each. If 4, 5 or 6 missing B0B0.

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1 1a k(16 9) + k(25 9) + k(36 9) (or 7k + 16k + 27k). M th = 1 M1 Þ k = 1 50 (answer given). * Model simple random variables as probability (3) 1b x P(X = x) Note: decimal values are 0.14, 0.32, 0.54 respectively th (5 marks) 1b Ignore any extra columns with 0 probability. Otherwise 1 for each. If 4, 5 or 6 missing B0B0. Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

2 2a = = 1 M1 4th = b P( 1 X < 2) = P( 1) + P(0) + P(1) = 0.6 4th 2c P(X > 2.3) = P( 2) + P( 1) + P(0) + P(1) + P = th (1) (1) (4 marks) Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

3 3a 2k + k k = 1 M th Þ 4k = 1, so k = 0.25 (answer given). * 3b P(X 1 + X 2 = 5) = P(X 1 = 3 and X 2 = 2) + P(X 1 = 2 and X 2 = 3) = = 0 (answer given). * 2.4 4th (1) 3c x1 + x P(X1 + X2) M th (3) 3d P(1.3 X 1 + X 2 3.2) = P(X 1 + X 2 = 2) + P(X 1 + X 2 = 3) M th = = or 5 16 ft (8 marks) 3b Must show that 5 can only be obtained from 2 and 3 or 3 and 2, and so must use P(X = 2) = 0 but condone explanation in words. 3c M1 for correct set of values for X 1 + X 2. Condone omission of 5 column. for correct probabilities for 0, 2 and 6. for others. Equivalent fractions are ,,,,, Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

4 4a Let X be the random variable the number of games Amir loses. X ~ B(9, 0.2) P(X = 3) = = to 3 sf from calculator 3.3 4b P( X 4) M th = awrt from calculator Use statistical tables and calculators to find cumulative (4 marks) 4a P( X 3) P ( X 2) = or 9! 3!6! (0.2)3 (0.8) 6 or 9 C or b 0.98 is M1A0 Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

5 5a X ~ B(20, 0.05) for for 20 and b 3.1b Understand the distribution (and its notation) and its use as a model. 5b P(X = 0) = (awrt) 3.4 5c P(X > 4) = 1 P( X 4) = = (2 s.f.) (answer given) * M th Use statistical tables and calculators to find cumulative (6 marks) 5b P(X = 0) = Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

6 6a X ~ B(15, 0.5) 3.1b for for 15 and b Understand the distribution (and its notation) and its use as a model. 6bi from calculator P(X = 8) = M bii P(X 4) = 1 P(X 3) = = awrt or M th Use statistical tables and calculators to find cumulative (6 marks) 6bi P(X = 8) = P(X 8) P(X 7) = or 15! 8!7! 0.58 (1-0.5) 7 or 15 C or = awrt or Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

7 7a 7a Binomial (distribution). 1.2 Each plate is either blue or not blue, independently of each other, with constant probability and there is a fixed number of them. 2.4 Understand the distribution (and its notation) and its use as a model. 7b X ~ B(10, 0.06) (could be seen in part a) 2.5 P(X > 2) = 1 P(X = from calculator M1 3.4 = awrt (3) (5 marks) Ignore any parameter values given for the first. For second all four points must be made with some context. Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

8 8a There is a fixed number of trials. 1.2 Each trial results in 1 of 2 outcomes, success and failure. 1.2 Probability of success on each trial is the same. 1.2 The trials are independent. 1.2 (4) Understand the distribution (and its notation) and its use as a model. 8bi P(X = 5) = 2 7 ; P(X 5) = 5 7 (either may be implied in either part) Idea of five failures followed by a success (or 0 successes out of five and then a success) seen or implied. 3.3 M1 3.4 P(5 on sixth throw) = M1 = awrt (5) 8bii æ B 8, 2 ö è ç 7ø or B æ 8, 5 ö è ç 7 ø seen or implied. M1 3.3 P(X = 3) = from calculator M1 (3) (12 marks) 8bii P(exactly 3 fives in first eight throws) = 8! æ 2ö 5!3! è ç 7ø 3 æ 5ö è ç 7ø 5 o.e. Education Ltd Copying permitted for purchasing institution only. This material is not copyright free.

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