‘Let's Make a Deal’ host Monty Hall has died
reuters.com
reuters.com
It's such a deceptively simple problem that's so easy to get wrong. I introduced it to my boss and he just wouldn't believe me when I told him the answer. He spent about 10 minutes formulating a simulation in Excel in order to convince himself.
I think the reason why problem is so unintuitive is some subtleties in how the problem is posed.
Monty does not just open a random door:
- He always opens one of the two doors you didn't choose.
- He knows where the prize is beforehand, and deliberately open a door which doesn't contain the price.
It should be stated clearly that these are the rules Monty follows. If the problem is stated like he just randomly decides to open one of the doors, then you can't say anything about how it affects the chances.
The citation on the wikipedia page does not really makes these rules explicit. It states
> the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat.
It is not really clear from this that the host are following the above stated rules. It is not stated explicitly that the host couldn't have chosen the door selected by the contestant, just that he didn't. If you don't know what rules he is following in selecting a door, then his door selection does not really give you more information. Stating the problem like this it is bordering on a trick question.
The really difficult thing to get your head around is that the door you chose still has a 1/3 probability, despite the fact that you know there are two doors, one of which hides the car, and the other hides the goat.
The selection process that went into choosing the other door makes it much more likely than the selection process for your current door.
RIP Mr. Hall.
> The show’s producers showed mercy on the “zonk” winners, however. After the taping of the show, they would be offered a substitute prize, such as a television, and most would take it. “In 1 percent of the cases, they didn‘t,” Hall said. “There was a time when a farmer won five calves and he wanted the calves. That cost me a fortune because when you rent them from the animal place, they’re expensive.”