

The Problem

 100 prisoners are individually numbered 1 to 100
 A room having a cupboard of 100 opaque drawers numbered 1 to 100, that cannot be seen from outside.
 Cards numbered 1 to 100 are placed randomly, one to a drawer, and the drawers all closed; at the start.
 Prisoners start outside the room
 They can decide some strategy before any enter the room.
 Prisoners enter the room one by one, can open a drawer, inspect the card number in the drawer, then close the drawer.
 A prisoner can open no more than 50 drawers.
 A prisoner tries to find his own number.
 A prisoner finding his own number is then held apart from the others.
 If all 100 prisoners find their own numbers then they will all be pardoned. If any don't then all sentences stand. 


The task

 Simulate several thousand instances of the game where the prisoners randomly open drawers
 Simulate several thousand instances of the game where the prisoners use the optimal strategy mentioned in the Wikipedia article, of:
 First opening the drawer whose outside number is his prisoner number.
 If the card within has his number then he succeeds otherwise he opens the drawer with the same number as that of the revealed card. (until he opens his maximum).


Show and compare the computed probabilities of success for the two strategies, here, on this page.




References

 The unbelievable solution to the 100 prisoner puzzle standupmaths (Video).
 wp:100 prisoners problem
 100 Prisoners Escape Puzzle DataGenetics.
 Random permutation statistics#One hundred prisoners on Wikipedia.


function shuffle(tbl)
  for i = #tbl, 2, -1 do
    local j = math.random(i)
    tbl[i], tbl[j] = tbl[j], tbl[i]
  end
  return tbl
end
 
function playOptimal()
    local secrets = {}
    for i=1,100 do
        secrets[i] = i
    end
    shuffle(secrets)
 
    for p=1,100 do
        local success = false
 
        local choice = p
        for i=1,50 do
            if secrets[choice] == p then
                success = true
                break
            end
            choice = secrets[choice]
        end
 
        if not success then
            return false
        end
    end
 
    return true
end
 
function playRandom()
    local secrets = {}
    for i=1,100 do
        secrets[i] = i
    end
    shuffle(secrets)
 
    for p=1,100 do
        local choices = {}
        for i=1,100 do
            choices[i] = i
        end
        shuffle(choices)
 
        local success = false
        for i=1,50 do
            if choices[i] == p then
                success = true
                break
            end
        end
 
        if not success then
            return false
        end
    end
 
    return true
end
 
function exec(n,play)
    local success = 0
    for i=1,n do
        if play() then
            success = success + 1
        end
    end
    return 100.0 * success / n
end
 
function main()
    local N = 1000000
    print("# of executions: "..N)
    print(string.format("Optimal play success rate: %f", exec(N, playOptimal)))
    print(string.format("Random play success rate: %f", exec(N, playRandom)))
end
 
main()