In the game of chuck-a-luck, a gambler chooses an integer from 1 to 6 and then three fair dice are rolled. If exactly \(k\) dice show the gambler's number then the payoff is \(k : 1\) for \(k \in \{1, 2, 3\}\); the bet loses if \(k = 0\). Random variable \(Y\) gives the number of dice that show the gambler's number and random variable \(W\) gives the gambler's net profit on a unit bet.