what if you chose the correct answer(purely by luck) on the first pick. Why switch?

Pointman, it's the probability of picking the correct one that the question is about.

You have no way of knowing whether you picked the correct one until the second door is opened. But your chances of winning increase if you switch. The key thing is that the presenter will **always** open a door with a goat behind it, regardless of whether you have picked the correct door in the first place or not.

Ryan, if I choose a door(say #1) and the host opens #3 with the goat, then there are 2 unknowns and the odds of me having chosen the correct door is 50%. Why is it to my advantage to switch? assuming I don't want the goat.

Pointman,

Say you choose door #1 - at this point you dont know which door the car is behind. you've made a random choice, selecting one out of 3 doors, so the probability that you have selected the correct door is 1 in 3, or 33%. Simple.

Since the probability must add up to 100%, this also means that the probability that the car is behind either door #2 or door #3 is 2 in 3, or 67%. Lets think of these doors as belonging to the host - so the host has a 2 in 3 chance of keeping his car.

Now to the important part. There are 2 key things:

1. The host

**knows** which door the prize is behind.

2. And,

**regardless of whether you initially selected the correct door or not**, the host will

**always** open one door with a goat behind it. In other words, the chances of him opening a door with a goat behind it are 100% - he is

**not** selecting at random.

Because of this, the probabilitty that the car was originally behind one of the host's doors remains unchanged (i.e. - it is still 67%)

Another way to think of itSometimes its easier to think of this problem, if we use 1000 doors rather than 3.

So - 1 car, 999 goats.

Same situation - you pick a door. The host then throws open 998 of "his" 999 doors, showing u goats behind all of dem.

Now, keeping in mind that he did

**not** select these doors at random - do you think it is more likely that the car is behind your 1 door (which you selected out of 1,000) or his one remaining door?