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# Discrete Probability

How often will a die come up "4"? How likely is it to rain tomorrow? Probability is one of the most powerful frameworks for modeling the world around us.

# Probability - By Outcomes

If two six-sided dice are rolled, the probability that they both show the same number can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

A fair 6-sided die is rolled twice. The probability that the second roll is strictly less than the first roll can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are positive, coprime integers. What is the value of $$a+b$$?

Details and assumptions

The roll of a dice refers to the value on the top face of the dice.

Two players each flip a fair coin. The probability that they get the same result can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

Lily has two bags containing balls. The first bag contains $$10$$ balls, with each ball labelled by a distinct number from $$1$$ through $$10$$. The second bag contains one ball labelled $$1$$, two balls labelled $$2$$, etc, up to ten balls labelled with $$10$$. Suppose Lily draws one ball from each bag uniformly at random and let $$\frac{a}{b}$$ be the probability that the two balls have the same value, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

If a 20-sided fair die with sides distinctly numbered 1 through 20 is rolled, the probability that the answer is a perfect square can be expressed as $$\frac{a}{b}$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a + b$$?

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