In a new twist on the classic Monty Hall problem, the optimal strategy reverses when the contestant secretly values a goat over the car. The puzzle, set earlier today, asks whether you should switch doors when you want to win a golden goat, not the car. The answer is no: you should stick with your original choice.
The Golden Goat Puzzle
In the standard Monty Hall problem, a contestant picks one of three doors. Behind one is a car, behind the other two are goats. After the initial choice, the host, who knows where the car is, opens a door revealing a goat, then offers the chance to switch. The well-known result is that switching doubles the odds of winning the car from 1/3 to 2/3.
In the new version, the contestant secretly knows that one goat is the former pet of an eccentric billionaire, worth more than the car. The host, unaware of this, opens a door revealing the ordinary goat. The contestant must decide whether to switch.
Why Sticking Wins
The solution uses frequencies. In 3,000 plays, the initial choice is the car in 1,000 games, in which case the host opens the ordinary goat 500 times (since he picks a goat at random). The initial choice is the ordinary goat in 1,000 games, but then the host never opens the ordinary goat (he would open the golden goat, but that is not the case here). The initial choice is the golden goat in 1,000 games, and the host opens the ordinary goat all 1,000 times.
Thus, in 3,000 trials, the host opens the ordinary goat 1,500 times. If the contestant sticks, they win the golden goat in 1,000 of those cases, a probability of 2/3. If they switch, they win in only 500 cases, a probability of 1/3. Sticking is better.
The puzzle, credited to Henk Tijms, was set by Alex Bellos, who has been publishing puzzles on alternate Mondays since 2015.
