Quantitative Finance
# Probability Theory

If 2 fair six-sided dice are tossed, what is the probability that they will show the same number?

Bret and Shawn are playing a game where each turn the players roll a 20-sided fair die and then add the number they get to their score. The winner is the player with the highest score at the end of the game (there may be a tie). Before the last turn, Shawn is ahead by 10 points. The probability that Bret can come back and win the game can be expressed as \( \frac{a}{b}\) where \(a\) and \(b\) are coprime numbers. What is \(a + b\)?

**Details and assumptions**

Both Shawn and Bret have 1 more roll on the last turn.

If there is a tie, no one won.

Two six-sided dice each have the numbers 1 through 6 on their faces. Neither die is fair, but they are both weighted the same. The probability of rolling a certain number on one die is given in the table below:

\(\begin{array}{c|cccccc} \mbox{number} & 1 & 2 & 3 & 4 & 5 & 6\\ \hline \mbox{probability} & \frac{1}{6} & \frac{1}{6} & \frac{1}{9} & ? & \frac{2}{9} & ?\\ \end{array}\)

If the probability that the two dice both show the same numbers is \(\left(\frac{2}{3}\right)^4\), we can express the probability of rolling 10 on these two dice as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

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