The random variable $$\(A\)$$ represents the score when a spinner is spun. The probability distribution for $$\(A\)$$ is given in the following table. (a) Show that $$\(\mathrm{E}(A)=3.5\)$$ (2) (b) Find $$\(\operatorname{Var}(A)\)$$ (3) The random variable $$\(B\)$$ represents the score on a 4 -sided die. The probability distribution for $$\(B\)$$ is given in the following table where $$\(k\)$$ is a positive integer. (c) Write down the name of the probability distribution of $$\(B\)$$. (1) (d) Given that $$\(\mathrm{E}(B)=\mathrm{E}(A)\)$$ state, giving a reason, the value of $$\(k\)$$. (1) The random variable $$\(X \sim \mathrm{N}\left(\mu, \sigma^{2}\right)\)$$ Sam and Tim are playing a game with the spinner and the die. They each spin the spinner once to obtain their value of $$\(A\)$$ and each roll the die once to obtain their value of $$\(B\)$$. Their value of $$\(A\)$$ is taken as their value of $$\(\mu\)$$ and their value of $$\(B\)$$ is taken as their value of $$\(\sigma\)$$. The person with the larger value of $$\(\mathrm{P}(X> 3.5)\)$$ is the winner. (e) Given that Sam obtained values of $$\(a=4\)$$ and $$\(b=3\)$$ and Tim obtained $$\(b=4\)$$ find, giving a reason, the probability that Tim wins. (2) (f) Find the largest value of $$\(\mathrm{P}(X> 3.5)\)$$ achievable in this game. (4) (g) Find the probability of achieving this value. (2)

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