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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.

# Level 3

There are 12 chameleons: 6 are blue, 6 are green.

They randomly form four groups of three. In any group, if there is only one chameleon of a particular color, it changes to the color of the other two. Otherwise, chameleons don't change color.

After this random grouping, if the probability that there will be nine blue chameleons and three green ones is $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, what is $$a+b$$?

###### Image credit: pixabay.com.

Three 6-sided fair dice are rolled together. Let $$P(n)$$ denotes the probability of obtaining a total sum of $$n$$. What is the relationship between $$P(9)$$ and $$P(10)$$?

Diana plays a special version of minesweeper on a square-shaped game field with $$n\times n$$ smaller squares, where $$n$$ is an integer greater than or equal to 4. First, she clicks any square, and then the game generates a mine in every non-clicked square, except one. Also, a number (an integer between 0 and 8) appears in the square Diana clicked, telling how many mines there are next to the square (in the small squares horizontally, vertically or diagonally touching it).

Diana's goal is to hit the only mineless square without hitting any mines before that. If Diana plays the best way she can, what is the probability she will succeed?

A box contains 4 balls. The color of each of the balls is one of the three: White, Black or Red. However, you don't know how many balls of each color are there. It might even be the case that all the balls are of the same color. You then draw two balls randomly from the box. The probability that both the drawn balls are red can be written as $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are coprime positive integers. What is the value of $$p+q$$?

The above picture is the bottom-left corner of a Minesweeper game. We are given that there are 4 total bombs among the 9 untouched squares (1 light blue and 8 blue squares excluding those squares with a red flag).

Find the probability that the highlighted (light blue) square is a bomb. Assume that each possible arrangement of bombs is equally likely (with "possible" meaning it satisfies the given number clues).

Notes: (in case you're unfamiliar with Minesweeper)

• The number on a white square represents the number of bombs adjacent to that square (including diagonally). These are squares which do not each have a bomb on them.

• A red flag indicates a square where we already know there is a bomb.

• Blue squares without a flag are "untouched," including the highlighted one. We are not given whether they have a bomb or not.

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