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No matter what they do each of the friends will have exactly 50% probability of answering correctly, this could lead some people to the incorrect conclusion that their probability of victory is 50% no … The theoretical optimal is that all three guesses are incorrect with 25% probability and exactly two of the guesses are correct with 75% probability. …

Three boxes labelled 1, 2 and 3 are placed in a room, each box has either a white or a black stone in it, with a 50% probability for each. … Variation 1 What is the optimal strategy if the stones are black with 70% probability and white with 30% probability? …

Hint: One could think that there would be a hard limit on the probability of scoring any trick at $5/6$, dropping to $4/5$, $3/4$, $2/3$ and $1/2$ on the last tricks, thus making the search for any strategy …

It is however given that the odds are the same for each roll, and that all 6 faces have a non-zero probability of coming up. … Create a scheme, using this die as the only source of random, that generates a number from 1 to 6 with a probability of exactly 1/6 for each possible outcome. …

I have designed a table of pairings for up to 6 rolls: Results after 3 rolls: ABC probability 4320/7776 Unpaired: AAA AAB ABA ABB Results after 4 rolls: AABB probability 540/7776 ABAB … probability 120/7776 AAABAC AAABCA AABAAC ABAAAC ABBBBC AAABCD probability 60/7776 AAABBC probability 120/7776 AAABCB AAABCC AABABC ABAABC ABBBAC AABABB probability 15/7776 ABAABB …

The anti-intellectuals have won the war, and under the iron rule of emperor Nonotin anyone who has been deemed too smart for their own good has been brought to labour camps. But most of the science f …