Mind and Body

As my body building friend has repeatedly yelled in the gym, training is about mind and body.



So to keep our minds sharp and healthy to stave of dementia, here is my puzzle thread where I will post a number of brain teasers.



Here is the first and second and third and nth (from a post I made on FB):

Quote:

Three prisoners sit in a circle with either a red or blue hat, at random, placed on their heads. They can not see their own hats or communicate to each other (they will be shot for violating the rules). They can only see the hats of their fellow prisoners They will be executed if at least one man does not guess the colour of his own hat. Every man has only the option of guessing their own hat colour or staying silent. If any man gets the colour of his hat wrong, they all die. There is no order for which they must say their guess or even an obligation to. And the warden kills all the prisoners if they don't guess within "n" minutes The crazy warden has allowed them to decide a strategy before the cruel game begins and before they are assigned a hat. Is there any strategy that gives the prisoners better than a 50% chance of execution... And now for the hard bit. With all the same conditions (red or blue hats and no communication once the game starts), the warden introduces fourth prisoner for the same cruel game and allows them to discuss strategy before hand. If they survive this, the warden will introduce a 5th and so on to the nth prisoner (keeping his promise not to execute them if they survive each game). EXTRA NOTE: This is based on logic. It's a math problem. You cant change the hats position (this is communicating) and each person recieves a particular color independently with probability 1/2. Is there an indefinite strategy that provides better than 50% chances of survival each round? (hint; yes there is)

Here is the probability of survival I've calculated for the n prisoners:

0.5+(1/(n^2))



If you can get higher, please post your working and I'll post mine.



THERE ARE NO LOOP HOLES, THIS IS A LOGIC PUZZLE.

Also if you answered this on FB correctly, please do not post your answer here unless you have calculated the nth.