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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 - Rule of Sum

Let $$X$$ be the number of televisions for a household chosen randomly. It is given that

\begin{align} P(X=1) & = 0.18, \\ P(X=2) & = 0.36, \\ P(X=3) & = 0.34, \\ P(X=4) & = 0.08, \\ P(X=5) & = 0.04. \end{align}

If $$P(X < 5) = a$$, what is the value of $$100a$$?

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 $$a + b$$?

In a health examination of the $$72$$ employees at Acme Corporation, the numbers of employees with blood type $$A$$ and $$B$$ were $$11$$ and $$19$$, respectively, and the remaining employees have blood type $$O$$. If one of the $$72$$ employees is selected uniformly at random, the probability that the person's blood type is $$A$$ or $$B$$ can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is $$a+b?$$

Raoul chooses an integer uniformly at random from $$1$$ to $$34.$$ The probability that his number is $$6$$ or $$32$$ can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b?$$

In a group of $$120$$ students, $$7$$ have black hair, $$35$$ have blond hair, $$12$$ have red hair, and the remaining have brown hair. If a student is selected uniformly at random from the group, the probability that the student has black hair or blond hair can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b?$$

Details and assumptions

Each student has exactly one hair color.

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