Don't feel bad. I'm in a similar boat. This one aggravated me all night.

If you do JoininNY's coin experiment the experiment proves the outcome.

The next issue, why?

It has nothing to do with Memory or Information or what you learn when the goat or penny is revealed.

On step one the chances of picking wrong are 2 out of 3.

On step two, if you do nothing at all the chances of being wrong remain the initial 2 out of 3

If you were dealing only with step 2 your chances of being wrong are 0.5

The overall chance of being wrong if you change your choice in step two is the product of 2/3 * .5 or 1/3 So by switching you change your chance of being wrong from 2/3 to 1/3.

It is totally counterintuitive but that is how probability works when you have a series of things happen each with its own probability.

The key to the puzzle is that by not switching you are retaining your initial high probability of having been wrong. By not switching you have taken only one toss of the dice not two.

Where you and I went wrong wasn't our initial view that there were only two choices behind each door (though that was part of it) it was our failure to recognize that by staying with the initial pick we had not tossed the dice twice. At least that is how I think we went wrong.

This statement by OPFOR isn't quite right al though I see what he is getting at.

"Paymeister, as has been said, you are correct in that any series of random events is random, and does not depend on preceding events (roulette spins, coin flips, etc). This, however, does NOT hold true here: there is only one event,;you are just given more information about it DURING the event."

There is ONLY one event ONLY if you DON'T switch. If you switch there are two events. And that is the key to making the probability end up in your favor. Switching creates two events so the probabilities ofan outcome from the two independent events can be multiplied. 2/3 * 1/2= 1/3 and the odds of your having picked wrong change from 2 in 3 to 1 in 3. I do not think the workings of this puzzle have anything to do with "you are given more information about it during the event." The revelation of where one penny or one goat happens to be doesn't in anyway change the odds or the necessary strategy. The key to the correct strategy is recognizing that you are being given a SECOND CHANCE with a second independent outcome from the first, and if you do nothing at all (don't switch) you are retaining the initial unfavorable odds.

You guys and my own slowness caused me a sleepless night. ACH!

Two quick points: 1) JoininNY said "do the experiment." He was right. That is how you get to see the truth.

2) JoininNY said this isn't a trivial matter. He is right again because we all make decisions with insufficient information all the time and often these

decisions are matters of guestimating the odds. If we don't come close, as some of us in this example, we can lose a lot of money, lose

an airplane, lose a war.