Envelopes of Money (Sept. 2024 challenge question)

Yeah, it makes no difference to you what's in the envelope the host takes, or even if he opens it. Your expectation is half the sum of the other two envelopes whether you swap or not.

Admin here...

This answer is correct. Thanks @Randall!

I would select the other envelope.The odds of getting the right one are 1 in 3 to start with.Now after one has been eliminated,the odds are 2 in 3 of getting it right, if you change doors.

It's similar to the game show on which door you chose #1, #2 and #3.

The odds still changes, whether the host knows or does not know what's in the envelope.

That only works if the host knows what the envelopes contain.

And of course, you have to assume that the host has an agenda.

The first one you picked had a 1 in 3 chance or being right.It still has a 1 in 3 chance of being right.

Now there are only 2 left,and your chances are 66% of getting it right if you change .

I admit,it is counterintuitive,but statistically that is how it works.

Since the host has no knowledge of which envelope is which, it is only by chance that he/she selected the one with $500. You now know that one of the remaining sealed envelopes has $100 and the other has $1000. You must choose 1 of 2, giving you a 50/50 chance.

Switch or not makes no difference! Randall's post #1 was correct.

"The first one you picked had a 1 in 3 chance of being right."

"Probability" has no concept of right: the envelopes might as well contain red, green and blue tokens; after you and the host choose envelopes, the probability of any token being in any envelope is the same for all three colours; when the host reveals what's in his envelope, the chances of one of the other colours being in any particular envelope is 50%.

The result can only change is an agency with a concept of value; knowledge of the positions, and, ability to change the result can have an effect.

When you selected your envelope, it was worth 1600/3 or $533.33. After you know that one of the other envelopes had $500, your expected return is $550.00. I would switch. Your odds will be the same or better. Whether the host knows which is which, he has still increased the odds with his selection.

Correction:

When you selected your envelope, it was worth 1600/3 or $533.33.

No, since the host draws the $500 envelope, your envelope contained either $100 or $1000, 50-50 chance, worth on average $550. Same with the other envelope.

The fact that the host drew his envelope after you is irrelevant. (Backward causality??)

Randall is correct...

If this isn't the Monty Hall problem, it's a close relative

Monty Hall Problem:

It's definitely NOT the Monty Hall problem linked in your post, because in this problem the host is not permitted to know the contents of the envelopes. This means the host cannot influence the outcome.

AND, in the Monty Hall problem linked in your post, switching doors is ONLY preferable IFF the host wants the contestant to win. If the host has been instructed to attempt to save the sponsor's money, then the contestant is better off NOT changing.

OK

"If the host has been instructed to attempt to save the sponsor's money, then the contestant is better off NOT changing."

It doesn't matter what the host wants, I assume he has to follow the rules set down by the shows producers. I don't think it has ever aired on this side of the pond, but, I assume that Monty never revealed a car after the contestant picked their first door. It would kind of spoil the shows ending if the contestant was offered the chance to choose the other closed door because he might get a better looking goat.

You switch envelopes. Marylin Savant answered this question several years ago and was railed on by many mathematicians. She subsequently proved them wrong and they, humbly, responded back that she was right and they were wrong. The answer of 2 chances to be wrong on the first pick and 50/50 on the second made switching the better decision. She also made a table of switching and not switching and switching was the best bet. Her IQ puts mine to shame so I'm going with the lady...

A link to her answer, please.

Your wish is my command :) Monty Hall problem - Wikipedia

Go back and read my post #10!

I'm sure it's me but I don't see the difference. He displays the contents of one of the unchosen envelopes. So, he may not have known the contents of any of the envelopes but he still revealed the contents of one of the unchosen envelopes. It might as well have been two envelopes with $500 and one with a $1,000. He has 1 chance in 3 of the $1,000 on the first go around and now has 1 in 2. The odds of not getting $1,000 the first time 2 out of 3. So it seems switching his first choice is the best bet. To me the difference in the two scenarios is superficial.

If I'm not mistaken that was for the Monty Hall problem. Apparently this is different, I haven't tried to think it through in detail.

With the exchange rate on the $ these days it matters little.

I think it is better not to switch. If you think about it from a Stress Perspective.

You are better to make a decision and then decide to be happy with that decision and don't think about it again. The more you think about it the more worry and the more stress and time wasted.

Once you have calculated that you only have a 50/50 chance, the odds are the same if you think about it for a long period or a short period. If you think about it for a short period you can apply your mind to something else, and maybe solve another issue which you can then benefit from. So you are better to go with your first pick and move on

Aesop's fable of the fox and the sour grapes has a similar concept

The easy way to understand the Monty Hall problem, is to imagine that instead of 3 doors there are 100: when the contestant chooses a door he effectively splits them into two groops: one of one door with 1 chance in 100 of hiding the car, and, one of 99 with 99 chances in 100 of hiding the car. Monty simply eliminates 98 goats from the second group. The wise choice then is to switch to the other group.