This variation tends to be counter-intuitive to those who understand the original problem. Many of them will insist it’s still 33/67.

You hare faced with 100 doors. Behind one of them is a car. You asked to pick one. After you pick one, the host open 98 doors without the car behind them leaving just the one you picked and one other.

You are now asked to stay with the one you originally chose or to switch. Clearly, if you played this game lots of times, you’d almost never do well by staying with your original choice. Only 1 in 100 times will that strategy pay off.

It’s exactly the same in the 3 door situation but since the numbers are less drastic, the intuition doesn’t work so well.

The last puzzle I have seen posed with a Ferrari as the prize. In explaining the counter intuative result you must take into account that there are two types of probability; prior and posterior. The later refers to the fact that the probability of an outcome changes when you have more information on the system. In this case the outcome is choosing the door with the Ferrari/car behind it.

Initially faced with 3 doors and 1 Ferrari, the prior probability of making the correct choice is 1/3. However if the games show hosts provides more information e.g. he tells one of the doors has a goat behind it , the odds change and become posterior probabilities. In this case they would become 50/50 without any choice being made or if he decided to tell you which door had the Ferrari then the odds would become better still.

So you choose your door and the game show host reveals a goat. You are now faced with two doors one of which has a Ferrari behind it. What information do you have that changes the odds from 50/50?

The answer is you know which door you chose first up. If you forget this, or the game show host decides to shuffle the remaining goat and Ferrari around, then the odds are back to 50/50. This is the intuitive situation that people tend to base their decision upon.

So you know which door you chose and you know that it has a 1/3 chance of having a Ferrari behind it. The other door obviously has a 2/3 chance. So make the switch and enjoy the drive………………unless you prefer goats.

But in Philip’s example, he states that the odds change, which they don’t. You pick one door, and you have a 1/3rd chance of being right. Even if they open all three doors at the same time, you still have 1/3rd chance of being right. But they don’t. They open 1 door and remove a door. Now, You are given a second choice, which the other door you didn’t pick has 50:50 chance (because, the door you have still is only 1/3 of being right, it’s frame of reference is based on the original deal). So, using that logic, you still should switch, because the other door is 50:50 (new deal) and your door is 33:66 (old deal)

RxIntern.

if you had a 1 in 3 chance of winning, and suddenly the odds get changed to a 1 in 2 chance of winning, switching will mean that you are working with 1 in 2 chances of winning, but choosing to switch or stay still makes it a 1 in 2 chance, the odds are already better but it is a 1 in 2 chance or 50% chance at winning not a 2/3 chance at winning.

The odds changed, and even if Marilyn vos Savant is a genius, this is one of those word play puzzles that confuses people but doesn’t prove anything.
So, from the standpoint of the beginning, the odds are 1 in 3. If one of the two possibilities you haven’t picked is removed, then you have a 1 in 2 chance of winning, therefore your odds of success have increased until you have a 1 in 2 or 50% chance of success.

“Just because the world agrees with false reasoning doesn’t make it right, it just makes the world wrong”.

-Steven Burda