2024 NJC P2 Q8

At a funfair game stall, players are allowed to choose two cards at random from six cards, with each card labelled with one letter from A to F. The player’s score, denoted by \(X\), is the Manhattan distance between the two squares corresponding to the player’s two chosen letters on the grid below,

where the Manhattan distance between two squares is the minimum total number of horizontal and vertical steps required to travel between them. For example, the Manhattan distance between B and F is \(2\), while the Manhattan distance between D and C is \(3\).

1. Tabulate the probability distribution of \(X\).[2]

The stall owner charges \(\$10\) per game and rewards the player with a cash prize, in dollars, of \(\frac{k}{10}\) times of the square of the player’s score, where \(k\) is a positive integer.

2. Determine the largest value of k for the stall to be profitable in the long run.[4]