A standard six-sided die is rolled repeatedly until the 4 is rolled. What is the probability that before the 4 was rolled all the odd numbers were also rolled at least once each?
pls help

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Two pairs of distinct points are chosen from the -by- grid below. What is the probability that the lines formed by both pairs are perpendicular?

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I would greatly appreciate solutions to this so I can check my work. Thanks! :-D

Hi, I am taking the AoPS Counting and Probability course. I am finding it difficult to solve some of them. Did any of you run into such issues and how did you move forward?

Sam had never fully understood the points system in soccer (football), and felt that the scoring of goals should be encouraged. His idea is that points should be awarded for a win, points for a draw and point for each goal scored, whatever the result of the match.

This was tried with three teams, Hubs, Dins and Rungs.
Each team scored at least one goal in every mach and no team played another more than once.
Hubs scored points, Dins points and Rungs scored points.

Find the score in each match.

[quote]Richard decides to read six books: the three books in The Lord of the Rings trilogy, as well as The Hunger Games trilogy. He must read the books of each trilogy in order, and he will not read a new book until he has finished reading the previous book. In how many different orders can Richard read these six books?[/quote]

This is probably just me being stupid, but I got this question wrong, and it said the answer was 72. Can somebody explain to me how?

A 100000 dollar treasure fell out of an airplane and landed in a swamp with a bomb that goes off in a day. Obviously, I want the treasure, so I hire mercenaries to find and defuse the bomb. If each squadron of mercenaries have a 90% chance of finding the bomb, and it costs 1000 dollars to hire a squadron for a day, how many squadrons should I hire to get the maximum profit? Assume that I am lazy and don't want to find the treasure myself, so if I hire no mercenaries there will be a 0% chance of getting the treasure back.

What is constructive counting in simple terms? The Intro to C&P book says "constructive counting is when we would construct the items we're trying to count": what does "construct the items" mean?

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books of the same kind are divided into groups. Each group has at least one book and each group has a different number of books. How many ways are there to divide the books?
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So I am going to start the Intro to CP book and i want to hear what you guys think about my question.
I want to the book alongside alcumus because im self studying without any class. So do you guys think its a better idea to read the chapters first and then do the alcumus or finish the whole book and then grind alcumus?