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A stopped clock gives the exact time twice a day, while a normally running (but out of sync) clock will not be right more than once over a period of months. A clever grandfather [as in grandfather clock] adjusted his clock to give the correct time at least twice a day, while running at the normal rate. Assuming he was not able to set it perfectly, how did he do it?
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Jacob and John are competing for the same girl. After years of battling, both decide to settle it by tossing a coin.
John produces a coin, but Jacob doesn’t happen to have one on him. Jacob is sure that the coin John has produced is loaded, i.e. it will come up with heads more than 50% of the time on average.
How do Jacob arrange a fair contest, based purely on chance and not skill, by flipping this coin?
Variation: (COIN BIASING) Jacob and John are competing for the same girl, and decide to settle it with a coin toss. John has known the girl longer than Jacob have, so Jacob agree that it is fair for him to have a chance of winning equal to P, where P > 0.5. However, Jacob only have a fair coin. How can you conduct this contest such that the biased probability is manifested? What is the average number of coin flips needed to determine a winner?
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A prisoner fate will be determined by a game. there are two jars, one with 100 white marbles, and one with 100 black marbles. at this point, prisoner is allowed to redistribute the marbles however he wish e.g. swap a black marble with a white marble, etc. the only requirement is that after prisoner is done with the redistribution, every marble must be in one of the two jars. Afterwards, both jars will be shaken up, and prisoner will be blindfolded and presented with one of the jars at random. then he pick one marble out of the jar given to him. if the marble prisoner pull out is white, prisoner live; if black, prisoner die. how should prisoner redistribute the marbles to maximise the probability that he live; what is this maximum probability (roughly)?
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