function rand(n) {
    return Math.floor(Math.random() * n);
}

function Game() {
    this.doors = ['closed', 'closed', 'closed'];
    this._goat = rand(3); // @private
}

Game.prototype.reveal = function(picked) {
    var options = [];
    for (var i = 0; i < 3; i++) {
        if (i !== picked && i !== this._goat) {
            options.push(i);
        }
    }
    this.doors[options[rand(options.length)]] = 'open';
    return this;
};

// Strategy prototype (abstract)

function Strategy() {
    this.picked = null;
}

Strategy.prototype.firstPick = function() {
    return this.picked = rand(3);
};

Strategy.prototype.finalPick = function() {
    throw new Error('unimplemented');
};


function HoldStrategy() {
}
HoldStrategy.prototype = Object.create(Strategy.prototype);

HoldStrategy.prototype.finalPick = function(game) {
    return this.picked;
};


function SwitchStrategy() {
}
SwitchStrategy.prototype = Object.create(Strategy.prototype);

SwitchStrategy.prototype.finalPick = function(game) {
    for (var i = 0; i < 3; i++) {
        if (i !== this.picked && game.doors[i] !== 'open') {
            return i;
        }
    }
    return null;
};


Game.monteCarlo = function(n, Strategy) {
    var total = 0;
    for (var i = 0; i < n; i++) {
        var game = new Game();
        var strategy = new Strategy();
        game.reveal(strategy.firstPick());
        total += (strategy.finalPick(game) === game._goat) ? 1 : 0;
    };
    return total / n;
};
Usage:
Game.monteCarlo(1000000, HoldStrategy);
// => 0.333622

Game.monteCarlo(1000000, SwitchStrategy);
// => 0.667626
More strategies are possible:
function CheatStrategy() {
}
CheatStrategy.prototype = Object.create(Strategy.prototype);

CheatStrategy.prototype.finalPick = function(game) {
    return game._goat;
};

Game.monteCarlo(1000000, CheatStrategy);
// => 1