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Answer 1

I'm not sure I fully understand the problem, but to start

each of the four coins has probability of 1/2 of landing heads

if there were two prisoners and just two coins the probability of both heads would be 1/2 times 1/2 = 1/4

for three prisoners, 3 heads probability would be (1/2)^3 = 1/2 times 1/2 times 1/2 = 1/8

for four prisoners: 4 heads' probability would be 1/16 = 1/2 x 1/8 = 1/2 x 0.125 = 0.0625 or 6 1/4 % chance

for N prisoners & N coins the probability of all heads is (1/2)^N = 1/2^N

I'm not sure what the point of a random number generator is, other than to confirm that their calculations are correct, run the experiment 100 times and they'd get 4 heads about 6+ times out of the 100.

That also doesn't take into account a logician prisoner being suicidal and refusing to flip the coin.

While unlikely, it is a real possibility and makes the likely probability of 4 heads = 0.0625 or less

Plus the guards may be dishonest and pretend the coins were not all heads when they were, further reducing the probability. Guards have been known to be sadistic, as in the Stanford prisoner experiment or American treatment of Muslim prisoners in Cuba.

Or if the guards really want to amuse themselves, they could pretend the coins came up all heads, and still torture the prisoners to death.

This all sounds similar to some California guards who did an office pool which prisoner would die first, after they intentionally placed gang member prisoners of opposite races in the same cell together. Do that 4 times and the probability of the one prisoner surviving 4 times is the same 1/16, ignoring other factors. Or like the movie Django, where a mandingo slave is up a tree with dogs ready to kill him, if he doesn't fight the 4th fight after winning 3 times.

All you can say is probability of 4 prisoners surviving is no greater than 6.25 %

Prisoners, even logicians, may miscalculate and be too disillusioned if the random number generator showed no heads after 100 flips. Prisoners in isolation too long may hallucinate and start to lose contact with reality.

Answer 2

My background is statistics rather than calculus, but I'll take a crack at solving it.

First, the logicians would understand that the probability of them escaping drastically decreases as more people flip coins. With the ideal number of people flipping being one. We get our probabilities using a binomial distribution.

P(success | 0 flippers) = (.5)⁰ = 0

P(success | 1 flippers) = (.5)¹ = .5

P(success | 2 flippers) = (.5)² = .25

P(success | 3 flippers) = (.5)³ = .125

P(success | 4 flippers) = (.5)⁴ = .0625

However, there is nothing to prevent them from all not flipping, this is where our random number generator comes in. Each logician will get a random number, and, if their output is below a cutoff (which we will set) they will decide to flip the coin.

The trick is now to set an optimal cutoff such that one person is likely to flip the coin, but reduces the chances of selecting no one. Let Θ denote the cutoff, interestingly, the probability of being under this cut off is also Θ, so we can model the number of flippers using a binomial.

(4 choose # flippers) (Θ)^#flippers(1-Θ)^{4 - #flippers}

We therefore want to find a Θ that maximizes the following joint probability:

(probability of success)x(probability of # flippers)...

(0) • (4 choose 0) (Θ)⁰(1-Θ)⁴ + (.5) • (4 choose 1) (Θ)¹(1-Θ)³ + (.25) • (4 choose 2) (Θ)²(1-Θ)²+ (.125) • (4 choose 3) (Θ)³(1-Θ)¹+ (.0625) • (4 choose 4) (Θ)⁴(1-Θ)⁰

To do this, I ran solver in excel on the above sum to find the most optimal cutoff (you could also set the derivative to zero but it's a little to early in the day for me to be going that). That being 0.342.

Unfortunately I'm unable to post a photo of my excel output on this post. However, this resulted in a 28.5% chance of our logicians being released.

Bonus!

I tried the above method using n = 20, this resulted in a cutoff of .069, and a total probability of success at 25.6%. Obviously this method doesn't scale well with N and there might be a mathematical method of exactly determining it. But I hope this helps as a start!

Answer 3

Since we are all logicians, we will all flip coins. So, our chances of release = 1/2 x 1/2 x 1/2 x 1/2 = 1 chance in 16.