State the axioms of probability.

State and prove Boole's inequality.

Suppose you toss a sequence of coins, the $i$-th of which comes up heads with probability $p_{i}$, where $\sum_{i=1}^{\infty} p_{i}<\infty$. Calculate the probability of the event that infinitely many heads occur.

Suppose you repeatedly and independently roll a pair of fair dice and each time record the sum of the dice. What is the probability that an outcome of 5 appears before an outcome of 7 ? Justify your answer.