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.

Able and Brainy are stumped by a multiple choice problem with three options and each of them takes a wild guess at it. When the proctor is distracted, they look at what answer the other has put down. Each of them has a fifty-fifty chance of either keeping his answer or copying his friend's. They get two more opportunities to crosscheck before they turn the test in. On both occasions, they behave exactly the same way, with the same probabilities.

What is the probability that they both have the correct answer at the end?

If the probability is in the form \( \frac mn\), where \(m\) and \(n\) are coprime positive integers, submit your answer as \(m+n\).

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 \(a+b\)?

Seven balls of different weights are randomly painted red, orange, yellow, green, blue, indigo and violet, each ball being painted a distinct color.

The green ball is found to be heavier than the blue ball, and the red ball is found to be heavier than the yellow ball.

Given just this information, if the probability that the red ball is heavier than the blue ball is \(\dfrac{a}{b}\), where \(a,b\) are coprime positive integers, find \(a + b\).

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