Answer posted tomorrow.
2 Probability Problems (3 pi each)
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Each person does a thumbs up if the other 2 have the same color of hat and a thumbs down if they have different colored hats. Then, they guess accordingly.
Mar 8, 2020
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#4
There are 8 possible combinations of hats, 1 combo is all blue, 1 is all red, 3 are one person red/two blue, and 3 are one person blue/two red. This means there is a 75% chance that they won't be all the same color. The best strategy has a 75% chance of being correct, because if this.
Here is the strategy:
If a person sees one of each color (on the other two players) they say "pass". If they see two red hats, they say "blue; and if they see two blue hats, they say "red".
No one will be wrong in 6/8 cases (75%) and only if all the hats are the same color will the players say something incorrect.
The answer and more math problems are here: https://www.math.ucdavis.edu/~gravner/MAT135A/resources/chpr.pdf
Where's my pi !?! LOL
5. From The New York Times, Science Times, D5, April 10, 2001:
“Three players enter a room and a red or blue hat is placed on each person’s head.
The color of each hat is determined by [an independent] coin toss. No communication
of any sort is allowed, except for an initial strategy session before the game begins.
Once they have had a chance to look at the other hats [but not their own], the
players must simultaneously guess the color of their own hats or pass. The puzzle
is to find a group strategy that maximizes the probability that at least one person
guesses correctly and no-one guesses incorrectly.”
The naive strategy would be for the group to agree that one person should guess and the
others pass. This would have probability 1/2 of success. Find a strategy with a greater chance
for success.
For a different problem, allow every one of n people to place an even bet on the color of his
hat. The bet can either be on red or on blue and the amount of each bet is arbitrary. The group
wins if their combined wins are strictly greater than their losses. Find, with proof, a strategy
with maximal winning probability.