Two dice are rolled. Find the probabilities of the following events.

a) the maximum of the two numbers rolled is less than or equal to 2;

b) the maxinum of the two numbers rolled is less than or equal to 3;

c) the maximum of the two numbers rolled is exactly equal to 3;

d) Repeat b) and c) with 3 replaced by \(x=1,…,6\);

e) Denote \( \mathbf{P}(x)\) as the probability that the maximum number is exactly \(x\).

Compute \( \sum_{x=1}^6\mathbf{P}(x)\).

[Pitman Page 10, #7]