Captain Haddock
- Location: Marlinspike Hall
- Registered: 8 Oct 2012
- Posts: 8,167
A letter in the FT is still 'doing my head in'! Enjoy.
Sir, I think that most of your readers must have heard of the Monty Hall problem many times (“Stick-or-switch inspires an onion of a puzzle”, Tim Harford, October 7). Here is a somewhat subtler puzzle for them to ponder on Saturday morning.
You are presented with two envelopes and told that one contains twice the amount of money than the other one. You pick one. The host opens the other one and shows that it contains $12. You can now change your mind and go with the opened envelope and win guaranteed $12 or you can remain with your first choice. In the latter case, the unopened envelope contains either $6 or $24 dollars with equal probabilities, so the expectation value is $15, which is $3 more than the guaranteed $12. So, should you change your mind or not?
The failure of the majority of people in understanding the Monty Hall and related problems is often taken as an illustration about the poor intuition humans have about probability theory. But humans per se also have a very poor intuition about numbers and can count to about five without any formal education. But speak to any 14-year-old and they have a pretty good understanding of what a thousand is.
It is for purely historical reasons that probability, combinatorics and statistics are relegated to optional courses taught at the end of secondary education, while we teach quadratic equations and basic trigonometry as basic maths. Taking probability theory as a core maths topic could reduce the number of problem gamblers and reduce the societal impact of terrorist attacks, in addition to drawing more children to mathematics with its truly puzzling puzzles.
Anže Slosar
London SE13, UK
"We are living in very strange times, and they are likely to get a lot stranger before we bottom out"
Dr. Hunter S Thompson
Reginald Barrington
- Location: Dover
- Registered: 17 Dec 2014
- Posts: 3,259
That is the Monty hall problem only with envelopes instead of doors!
and always swap.
Arte et Marte
Captain Haddock- Location: Marlinspike Hall
- Registered: 8 Oct 2012
- Posts: 8,167
I disagree Reg. I am convinced that the chances in the Monty Hall problem change when the first goat is revealed and have run computer simulations and drawn probability-space diagrams to prove it.
In this case above I think swapping makes no difference. For example IF you opened YOUR first choice envelope rather than the other and found the 12 would you 'swap'. The 'fact' that the average is 15 suggests you should BUT do we take into account that you forgo your original choice of 12?
"We are living in very strange times, and they are likely to get a lot stranger before we bottom out"
Dr. Hunter S Thompson
Reginald Barrington- Location: Dover
- Registered: 17 Dec 2014
- Posts: 3,259
But the unopened envelope is the equivalent of the 2 unopened doors, substitute the goats in the Monty problem with the 6, 12 and 24 dollars or vice versa, Though with bigger envelopes.
Arte et Marte
Captain Haddock- Location: Marlinspike Hall
- Registered: 8 Oct 2012
- Posts: 8,167
A further thought Reg. I think the difference is the sort of information divulged by the opening of the door/envelope.
In classic Monty Hall the goat revelation means a losing position which informs the change of chance in the game?
In the above the 12 has no status as a loss or win UNTIL we find what's in the other envelope. I think.
"We are living in very strange times, and they are likely to get a lot stranger before we bottom out"
Dr. Hunter S Thompson
Reginald Barrington- Location: Dover
- Registered: 17 Dec 2014
- Posts: 3,259
I think you're right it is indeed a schrodinger cat paradox, the divulging of the 12 dollars is irrevelant.
Arte et Marte
Captain Haddock- Location: Marlinspike Hall
- Registered: 8 Oct 2012
- Posts: 8,167
I too THINK I'm right. The annoying thing is I can't yet convince myself I'm right!
"We are living in very strange times, and they are likely to get a lot stranger before we bottom out"
Dr. Hunter S Thompson
Reginald Barrington- Location: Dover
- Registered: 17 Dec 2014
- Posts: 3,259
https://en.m.wikipedia.org/wiki/Two_envelopes_problem
I'm going to go and have a little lie down, let me know if you convince yourself!
Arte et Marte