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Probability theory is very nice. It contains many questions which are very easy to state, but not so easily solved. Let's see if you can solve these questions.

Here you go:

Thank you all for participating! I hope many of you have fun with this! Don't hesitate to post any feedback in the thread!

More information:

- For an answer to count, not only the answer must be given but also a detailed explanation.
- Any use of outside sources is allowed, but do not look up the question directly. For example, it is ok to go check probability books, but it is not allowed to google the exact question.
- If you previously encountered this statement and remember the solution, then you cannot participate in this particular statement.
- All mathematical methods are allowed.
- Please reference every source you use.

Here you go:

- SOLVED BY BiGyElLoWhAt, PeroK The universe is about ##432,043,200,000,000,000## seconds old. Assume that from the very start of the universe, a solitary monk was flipping a coin over and over again. He flips a coin every second. A run of order ##n## are ##n## consecutive heads in a row, or tails in a row. On average, what is the largest run that the monk has had?

- SOLVED BY ProfuselyQuarky Every cereal box contains one toy. There are in total ##20## different toys. How much boxes of cereal do you expect to buy - on average - in order to collect all the toys.

- REMOVED

- SOLVED BY PeroK A closet contains ##100## pairs of shoes. I choose at random ##8## shoes. What is the probability that I will have selected exactly one complete pair?

- SOLVED BY mfb In a town of ##n+1## inhabitants. A person tells a rumor to two distinct numbers, the "first generation". These repeat the performance and generally, each person tells a rumor to two people at random without regard to the past development. Find the probability that the generation ##r## will not contain the person who started the rumor. Find the median of this distribution assuming ##n## large.

- SOLVED BY andrewkirk Find the probability that in a random arrangment of ##52## cards no two aces are adjacent.

- SOLVED BY Fightfish, mfb In a ballot, candidate Trump scores ##304## votes and Clinton scores ##216## votes. When the votes are counted, find the probability that throughout the counting, there are always more votes for Trump than Clinton.

- SOLVED BY thephystudent I have a stick of ##1## meter. I break this stick in ##3## pieces in such a way that every point of the stick has as much chance of being a break point. Find the probability that I can combine the three pieces into an obtuse triangle.

- SOLVED BY Charles Link, PeroK In a room of ##1000## people. How many people do you expect there to be so that nobody else in the room shares their birthday?

- REMOVED

Thank you all for participating! I hope many of you have fun with this! Don't hesitate to post any feedback in the thread!

More information:

- gato-docs.its.txstate.edu/mathworks/DistributionOfLongestRun.pdf
- http://www.math.uah.edu/stat/urn/Coupon.html

- REMOVED
- Feller "An introduction to probability theory and its applications Vol1" Chapter II "Elements of Combinatorial Analysis" Exercise 26
- Feller "An introduction to probability theory and its applications Vol1" Chapter II "Elements of Combinatorial Analysis"
- Feller "An introduction to probability theory and its applications Vol1" Chapter II "Elements of Combinatorial Analysis" Exercise 42

Solution: Look at ##48## non-aces. These determine ##49## gaps. Choose ##4## gaps. - Feller "An introduction to probability theory and its applications Vol1" Chapter III "Fluctuations in coin tossing and random walks"
- http://www.math.uah.edu/stat/buffon/Triangles.html
- Invented on my own
- REMOVED

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