In a game of chess, a player can either win, draw or lose. The probability that Nihan wins any game of chess is 0.5. The probability that Nihan draws any game of chess is 0.3. Nihan plays 2 games of chess.

1. In a game of chess, a player can either win, draw or lose. The probability that Nihan wins any game of chess is 0.5. The probability that Nihan draws any game of chess is 0.3. Nihan plays 2 games of chess. 

a) Construct a tree diagram to show the above information. (5 marks)

b) Work out the probability that Nihan will win both games? (2 marks)

c) Nihan’s friend, Kaisen, said ‘the probability she does not drawin games of chessis 0.39. Is he correct?Explain your answer. (2 marks)

2. 

a) On a menu in a restaurant there are 6 starters, 5 main courses and 3 desserts to choose from.  

i. How many different three-course meals could you have?                               (1 mark)

ii. In a two-course meal one of the meals has to be a dessert. How many different two-course meals could you have? (1 mark)

b) Yujie has two fair spinners. Spinner A has three equal sections— 2 black and 1 yellow. Spinner B has five equal sections¬¬— 3 black and 2 yellow. He spins spinner A, then spinner B. 

iii. How many possible outcomes are there? (1 mark)

iv. What is the probability exactly one lands on yellow?          (2 marks)

The random variable Y has mean 2 and standard deviation of 3.Find:

Hint: Express these in terms of the original E(Y) and Var(Y)

Var (Y)= (1 mark)

E[3Y+1]= (2 marks)

Var[3Y+1]=(2 marks)

E[Y2]= (2 marks)

E[(Y-1)(Y+1) ]= (3 marks)

Var[(Y-1)/3]=(2 marks)

The scatter diagram below illustrates the overall lengths l metres and the typical operational speeds sknots (nautical miles per hour) of 12 container ships. The length of a ship is one of the factors which determines its typical operational speed. 

Summary statistics for these data are as follows:

n=12 Σl=2219Σs=234.6Σl^2=443867

Σs^2=4700.56Σls=45149.0

Calculate the equation of the linear regression line of son l.              (7 marks)

Calculate the product moment correlation coefficient (r).(3 marks)

The coefficient of determination r^2=81.5%. Give an appropriate comment.(1 mark)

The equation of the new regression line is:

s = 0.0453l + 11.5

Calculate the estimate for an overall length of 100 metres using this new equation.(2 marks)

5. A fair cubical (six-sided) die is thrown four times.Use the binomial probability formula to calculate:

a) The probability of no threes.                                                                          (2 marks)

b) The probability of at least 2threes.(7 marks)

6. A college has 3 sports clubs: football, rugby, and basketball. Michael is conducting a stratified sample of sports players at his college. The table below 

gives some information about the sizes of the groups.

Sport Football Rugby Basketball

Number of players 196 91

Number in sample 28 19

a) Complete the table.                                (3 marks)

b) A Volleyball sport club has been added to this college. The number of Volleyball players are 120. A new stratified sample of size 60 is required. Calculate 

the number of each type of sport club that should be chosen.(3 marks)

7. TherandomvariableXrepresentstheweight,ingrams,ofachocolatebar. Itisknownthat X is Normally distributed with mean 50.7 and variance 0.72. On the wrapper it states that the bar weighs 50 grams.

a) Find the proportion of these chocolate bars that weigh at least50 grams.  (3 marks)

b) A quality control manager wishes to increase this proportion to95%.

i. Find the required value of the mean if the varianceremainsunchanged.(4 marks)

ii. Find the required value of the variance if the meanremainsunchanged.(2 marks)

8. The weights of another type of chocolate bar are also Normally distributed. On the wrapper it states that the bar weighs 25 grams. It is known that 99% of these bars weigh at least 25.0 grams and 75% of them weigh at least 25.4 grams.

a) Find the probability that one of these bars weighs at least26.0 grams.(6 marks)

b) One bar of the first type (from Q7) and 2 bars of the second type (Q8) are selected at random. Find the probability that at least one of the bars has a weight less than that stated onits wrapper.(2 marks)

9. The time series graph gives information about the total amount of money spent by overseas tourists in the UK for each quarter for the years 2011 to 2013.

A trend line has been drawn on the graph. The trend line is based on 4-point moving averages.

a) Explain why 4-point moving averages were chosen.(1 mark)

b) Discuss any seasonal variation shown by thegraph. Do not do anycalculations. (2 marks)

c) i.  Work out the gradient of the trend line.                                         (2 marks)

ii. Hence, Interpret your answer.                                                              (1 mark)

d) Calculate an estimate for the total amount of money spent by overseas tourists for Quarter 1 in 2014. You must show yourworking.(3 marks)

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