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If we're having a thread about probability theory, then we must have one on statistics too! The following questions are all very open-ended and thus multiple answers may seem possible. Your goal is to find a strategy to find the answer to the questions. Furthermore, you must provide some kind of reasoning as to why your strategy is a decent one.

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 strategy. An explanation of why the strategy is plausible must be given. Don't forget the detail the model you're working with and why this model is plausible.
- 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.
- What I feel are the best answers will be awarded on this original post. If you think you came up with a better answer, you must prove your answer is better.

- SOLVED BY mfbEvery day there is one train between Mordor and Rohan. These are the number of people who want to take the train on one day:

Day 1: 233

Day 2: 231

Day 3: 254

Day 4: 212

Day 5: 202

Find the optimal number of seats in the train.

- Take the following two sequences of coin tosses:

Code:`THHHHTTTTHHHHTHHHHHHHHTTTHHTTHHHHHTTTTTTHHTHHTHHHTTTHTTHHHHTHTTHTTTHHTTTTHHHHHHTTTHHTTHHHTHHHHHTTTTTHTTTHHTTHTTHHTTTHHTTTHHTHHTHHTTTTTHHTHHHHHHTHTHTTHTHTTHHHTTHHTHTHHHHHHHHTTHTTHHHTHHTTHTTTTTTHHHTHHH`

Code:`THTHTTTHTTTTHTHTTTHTTHHHTHHTHTHTHTTTTHHTTHHTTHHHTHHHTTHHHTTTHHHTHHHHTTTHTHTHHHHTHTTTHHHTHHTHTTTHTHHHTHHHHTTHTHHTHHHTTTHTHHHTHHTTTHHHTTTTHHHTHTHHHHTHTTHHTTTTHTHTHTTHTHHTTHTTTHTTTTHHHHTHTHHHTTHHHHHTHHH`

- SOLVED BY Math_QED I want to estimate the number of fish in a lake. I catch 400 fish and given them all a red dot. I throw them back in the lake. Then I catch 400 fish again. I note that 100 of them have a red dot. How many fish are there in the lake?

- SOLVED BY Ygggdrasil, mfb Unstable particles are emitted from a source and decay at a distance ##x##, a real number that has an exponential distribution with characteristic length ##\lambda##. Decay events can be observed only if they occur in a window extending from ##x=1## to ##x=20##. We observe ##6## decays at locations ##\{2,5,12,13,13,16\}##. What is ##\lambda##?

- SOLVED BY mrspeedybob, PeroK I have a big box filled with balls. All balls have a number. I draw ##5## balls at random and record their number. They are: ##10##, ##50##, ##104##, ##130##, ##213##. How many balls do you expect to be in the box?

- I claim I can tell the difference between coca cola and pepsi cola better than just guessing. Somebody pours 5 cups of pepsi and 5 cups of coca cola and hands them to me. I tell them which cup is which. It turns out I judged correctly 8 of the 10 cups. Do you believe my original claim?

- SOLVED BY MarneMath A professor got a ticket twelve times for illegal overnight parking. All twelve tickets were given either Tuesdays or Thursdays. Is it justified for him to rent a garage on these days?

- SOLVED BY QuantumQuest In a certain family, four girls take turns at washing dishes. There were four breakages. Three of them were caused by the youngest girl. Is it justified to call her clumsy?

- SOLVED BY fresh_42 Given the following encoded text, find out whether this is a real text or randomly generated using some scheme. Attempting to decode the text doesn't count.

Code:`sedwhqjkbqjyedi oek xqlu tusetut jxyi junj jxqj mqi dej fqhj ev jxu fherbuc jxekwx ie oek wuj de feydji jxu vebbemydw yi qd unsuhfj vhec myayfutyq fherqrybyjo yi jxu cuqikhu ev jxu byaubyxeet jxqj qd uludj mybb esskh fherqrybyjo yi gkqdjyvyut qi q dkcruh rujmuud puhe qdt edu mxuhu puhe ydtysqjui ycfeiiyrybyjo qdt edu ydtysqjui suhjqydjo jxu xywxuh jxu fherqrybyjo ev qd uludj jxu cehu suhjqyd mu qhu jxqj jxu uludj mybb esskh q iycfbu unqcfbu yi jxu jeiiydw ev q vqyh kdryqiut seyd iydsu jxu seyd yi kdryqiut jxu jme ekjsecui xuqt qdt jqyb qhu ugkqbbo fherqrbu jxu fherqrybyjo ev xuqt ugkqbi jxu fherqrybyjo ev jqyb iydsu de ejxuh ekjsecu yi feiiyrbu jxu fherqrybyjo yi edu xqbv eh vyvjo fuhsudj ev uyjxuh xuqt eh jqyb yd ejxuh mehti jxu fherqrybyjo ev xuqt yi edu ekj ev jme ekjsecui qdt jxu fherqrybyjo ev jqyb yi qbie edu ekj ev jme ekjsecui jxuiu sedsufji xqlu ruud wylud qd qnyecqjys cqjxucqjysqb vehcqbypqjyed yd fherqrybyjo jxueho iuu fherqrybyjo qnyeci mxysx yi kiut mytubo yd iksx qhuqi ev ijkto qi cqjxucqjysi ijqjyijysi vydqdsu wqcrbydw isyudsu yd fqhjyskbqh fxoiysi qhjyvysyqb ydjubbywudsu cqsxydu buqhdydw secfkjuh isyudsu wqcu jxueho qdt fxybeiefxo je veh unqcfbu thqm ydvuhudsui qrekj jxu unfusjut vhugkudso ev uludji fherqrybyjo jxueho yi qbie kiut je tuishyru jxu kdtuhboydw cusxqdysi qdt huwkbqhyjyui ev secfbun ioijuci`

- SOLVED BY jbriggs444 A person is playing a game operated by a psychic, an entity presented as somehow being exceptionally skilled at predicting people's actions. It is known that the Psychic predicts people's actions correctly in approximately 99.9% of the cases. The player of the game is presented with two boxes, one transparent (labeled A) and the other opaque (labeled B). The player is permitted to take the contents of both boxes, or just the opaque box B. Box A contains a visible $1,000. The contents of box B, however, are determined as follows: At some point before the start of the game, the Psychic makes a prediction as to whether the player of the game will take just box B, or both boxes. If the Psychic predicts that both boxes will be taken, then box B will contain nothing. If the Psychic predicts that only box B will be taken, then box B will contain $1,000,000.

If the psychic predicts that the player will choose randomly, then box B will contain nothing.

By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined. That is, box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Psychic is powerless to change the contents of the boxes. Before the game begins, the player is aware of all the rules of the game, including the two possible contents of box B, the fact that its contents are based on the Psychic's prediction, and knowledge of the Psychic's infallibility. The only information withheld from the player is what prediction the Psychic made, and thus what the contents of box B are.[/COLOR]

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:

- Invented myself
- gato-docs.its.txstate.edu/mathworks/DistributionOfLongestRun.pdf
- Feller "An introduction to probability theory and its applications Vol1" Chapter II "Elements of Combinatorial analysis"
- MacKay "Information Theory, Inference and Learning algorithms" http://www.inference.phy.cam.ac.uk/itila/p0.html
- http://www.math.uah.edu/stat/urn/OrderStatistics.html
- https://en.wikipedia.org/wiki/Fisher's_exact_test
- 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"
- https://en.wikipedia.org/wiki/Newcomb's_paradox

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