1. A game is based on drawing two cards from a modified pack. The pack is a reduced set with cards from Ace to Ten and the cards are laid in two piles, red and black. A card is chosen at random from each of the piles and the scores are added. Rewards are given if the score is divisible by three (‘making a three’) or by five (‘making a five’), and not given otherwise.

Count the number of ways a score of ten can be produced. [1]

Show that there are 20 ways of scoring a multiple of five and demonstrate how many ways there are of ‘making a three’. [4]

Write the probability distribution for the (three or four) classes of result. [3]

The game costs a stake of ¥3 for each turn. The rewards are given by the formula ¥3 for a ‘three’ and ¥5 for a ‘five’ and ¥20 for a fifteen. Calculate the expected reward from the game. [5]

2. A business manufactures light bulbs. The process engineer is worried that the products made are faulty. Each process machine produces 900 light bulbs an hour. Samples of twenty bulbs are taken every quarter-hour and tested to see if they work at all.

After running the tests for ten hours, for the figures for failures are as follows:

1 0 2 3 2 1 0 0 1 0 1 1 1 1 0 1 2 1 2 1

2 1 0 3 0 3 0 1 0 0 1 0 2 1 1 0 1 3 4 0

Copy and complete frequency table for this data: x 0 1 2 3 4 5 6

N 1 0 0

What is the most likely number of failures in a batch of twenty? What is the expected number of failures in a batch? What is the variance for the samples? [6]

Extend the result from the previous part to estimate the probability that a bulb picked at random is faulty. Predict an acceptable numbers of failures produced by any single machine in an hour. Comment on the choice of a suitable distribution to model this situation. [4]

3. A farmer has been offered a new feed for his cows. The supplier says that the new feed is preferred by five out of eight cows. The farmer accepts the claim, but wants to test it. He sets up an experiment so that a few of the feeding stalls offer the cows a choice between the new and the old.

a) If there are three feeding sessions a day, what is the probability that the same choice is made three times? [3]

He sets up a simple counting device to see which is used more often. After a hundred days he should have 300 choices made. [A hundred days is how long each cow will give milk; he will want to change the feed for milking and non-milking cows]

b) How many times would he expect the new feed to have been chosen? If he thinks the cows have not chosen as expected: what range of numbers (choosing the new feed) would he expect to find so as to have only a 5% chance of making a wrong assumption in his conclusion? What other assumptions has he made? [1,2,2]

4.The masses of 16 apples have been measured and the data collected are as follows:

23.5, 24.7, 25.2, 28.1, 29.9, 27.5, 23.6, 25.3, 27.0, 21.8, 22.7, 26.2, 30.6, 12.1, 28.9, 30.1

a) produce a box plot (or a box and whisker diagram) for the data. [4]

b) Hence establish some bounds for outliers and mark them on your diagram. [3]

Another sample is such that n = 12, x = 282 and x2 = 6838.68

c) Find the mean and standard deviation for this second sample. [3]

d) Using the information you have generated, possibly using a diagram, comment whether you think the two samples are from the same orchard. [2]

DJS 20081115

...and the front page looked like this....

UNIVERSITY OF CAMBRIDGE UNINTENTIONAL EXAMINATIONS

General Certificate of Education

Advanced Subsidiary Level and Advanced Level

MATHEMATICS A2 9709/05

Paper 5 Sadistics 1 retest (S1) Noremember 2008

Up to 1.5 hours

Additional Materials:Answer Booklet/Paper

READ THESE INSTRUCTIONS FIRST

If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your English Name on all the work you hand in.

Write in dark blue or black pen.

You may use a soft pencil for any diagrams or graphs.

Do not use staples, paper clips, highlighters, glue or correction fluid (all of these are forbidden examination materials).

Make sure your phone is turned off.

Make sure anything that could be described as notes is well out of reach.

Put any drink out of sight and leave it there. Check that you are wearing your uniform properly.

Answer all the questions.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.

The use of an electronic calculator is expected, where appropriate.

All formulae are assumed to be known.

You are reminded of the need for clear presentation in your answers.

At the end of the test, hand in ALL the papers, both the question paper and your worked answers.

Make sure your name is on every answer sheet.

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 50.

Q1 9 ways of making a ten. 20 of a five (including the 15). Three or four ways because you want to sapearate out the 15s from the 3s and 5s. Out of 100 ways (some students will find 400), 53 are neither. The reward is -29p (from memory)

Q2 E(X) = 9/8, Var (X) = 1.11 45 failures out of 400 sampled. Use µ±2ø for typical values. Normal. FM students can correctly say Poisson, but not known to S1 students. Predictable question.

Q3 Many students miss the two cases 5/8 cubed and 3/8 cubed. 187.5 = E(X) Use µ±2ø again. Assumptions include cows acting independently of each other. Some studdents will insist that 5/8 should be 1/2 throughout. Treat as mis-read.

Q4 Boxplot requires median, quartiles, etc so outliers defined by 1.5(IQR) method. Use µ±2ø in second half only. (d) could be S2 or FM but does not require a test, only an opinion; the likely upper limits match but the evidence is largely inconclusive - not extreme enough to say they are from different orchards.

Objectives: to stretch the language; to emphasise various points not well grasped.