[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

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Each card in a deck of cards bear a single number from 1 to 5 such that there are $n$ cards bearing the number $n$, where $n = 1, 2, 3, 4, 5$. One card is randomly drawn from the deck. Let $X$ be the number on the card drawn.

(i) Find the probability distribution of $X$.

(ii) Show that $\mathbb{E}(X) = \frac{11}{3}$ and find $\text{Var}(X)$.

Andrew draws one card from the deck, notes the number and replaces it. The deck is shuffled and Beth also draws on card from the deck and notes the number. Andrew’s score is $k$ times the number on teh card he draws, while Beth’s score is the square of the number on the card she draws. Find the value of $k$ so that the game is a fair one.