Probability Puzzles

Jacob & John have two children. The probability that the first child is a girl is 50%. The probability that the second child is a girl is also 50%. Jacob & John tell you that they have a daughter.
What is the probability that their other child is also a girl?
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A jar contains one hundred marbles, each of which may be white or black. You pull out 100 marbles with replacement, and they are all white. What is the probability that all one hundred marbles are white?
(Note: “With replacement” means you take out a random marble, look at its colour, then put that marble back. Then repeat.)
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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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Three logicians went to bar for having a drink.
The bar person asks the logicians “Do they all want to have the drink?”.
The first logician replied, ” I don’t know”.
The second logician replied, ” I don’t know”.What is the reply of the last logician?
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What if somebody offered to bet that at least two people in your Social group had the same birthday?
Would you take the bet?
How large must a social group be to make the probability of finding two people with the same birthday at least 50%?
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At the local Puzzlefry Internet dispatch center, the packing department was getting ready for the weekend.
However, it was late in the day and the final five packages were ready to be labelled.
Unfortunately, John dropped the labels to the packages and then had no idea which label belonged to which packet.
What is the probability that John managed to correctly label exactly four of the five packages?
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