Description

In the game of high-low two fair dice are rolled. The gambler can make three different bets based on the sum \(Y\) of the dice scores \(X_1\) and \(X_2\):

1. Bet low: This bet pays \(1 : 1\) if \(Y \in \{2, 3, 4, 5, 6\}\) and loses otherwise.
2. Bet 7: This bet pays \(4 : 1\) if \(Y = 7\) and loses otherwise.
3. Bet high: This bet pays \(1 : 1\) if \(Y \in \{8, 9, 10, 11, 12\}\) and loses otherwise.

Random variable \(W\) denotes the net winnings on a unit bet.