Door to door challenge.

The answer is correct

Here's my explanation, which can't be that good but we don't have the winning explanation.

Hopefully it will go some way to show you why you have better odds of winning if you swap doors.

Lets assume for this discussion that we initially pick door number 1, and then look at the other possible scenarios at what might be behind the other 2 doors, which I've tried to show below in the table.
On the 1st row, if you happen to have picked the prize, then by switching you would loose.
On the 2nd row, you picked an empty door, the host has shown you an empty door, therefore switching would mean you win
On the 3rd row, you picked an empty door, the host has shown you an empty door, therefore switching would mean you win
So, this means by switching you have a better chance of winning the prize.
Staying with the same door gives a 1 in 3 chance of winning.
Switching gives a 2 in 3 chance of winning.

 

The problem is called the Monty Hall Problem after the American game show hosted by Monty Hall that featured this sort of decision.

The explanation in the newsletter is somewhat unconvincing. However there is a watertight mathematical proof that switching gives you the best chance of winning.

The usual explanation is assume the same problem was given with a million doors. Now after you select a door the host opens 999,998 doors showing nothing leaving you with the same 1 door you chose vs 1 door the host left. Should you switch? Your first choice was 1 in a million to have picked the prize, if you switch you are picking the door that has 999,998 chances in a million of being right.

Note there are lots of variants of this by which switching is NOT the best solution. The reason switching IS the best answer for this problem is that the host KNOWS WHERE THE PRIZE IS and thus opens doors where the prize is not thus leaving a choice between prize and no prize.

Thus the odds are say 1 in a million you chose right first time vs the door that the host was forced to leave after showing the other wrong answers. This door the host was forced to leave has 999,998 chances in a million of having a prize with a million doors or a 2 in 3 chance with just 3 doors.

 

The aim of the challenge was to improve readers' understanding of probability so that they could begin to apply it to the problems that they are faced with in their research.

The key insight is that the host opening the door makes no difference whatsoever. Whilst many entrants submitted tables or provided examples where there were extra doors, I didn't feel that either of those approaches would help sceptics to understand why their solution was incorrect. The fact is that by switching the contestant gives herself the same chance as if she had chosen two doors in the first place, so changing the problem around seemed to be the best way of making that clear.

However I didn't expect everyone to be convinced by the solution I provided - when this problem featured in New Scientist in 1997 it provoked heated debate (as it has on many other occasions). The explanation given then was: "There is a 1/3 chance of the £10 000 being behind door A, and a 2/3 chance of it being behind B and C together. If you know there is a zero chance of it being behind C, then there must be a 2/3 chance of the money being behind B. Thus, switching doubles your chances."

I'm not surprised that explanation didn't convince everyone either!

 

One of the key things for skeptics to realise is that the problem is very different from games like "Deal or No Deal" on Channel 4 where at the end of the game there is sometimes the offer to "swap boxes". This is a completely different problem. In the deal or no deal instance no-one involved in the decision knows what is in either box so the "reveal of a wrong box" that is at the core of the Monty Hall problem simply doesn't apply.

In the deal or no deal scenario it is a genuine 50-50 chance, in the Monty Hall problem its a 2/3rds vs 1/3rd chance. The difference is in the elimination in Monty Hall being based on knowledge.

 

He was talking about statistics that are prepared by other people. What I was talking about in my newsletter was generating your own statistics in order to make more sense of what you are seeing. There's a very big difference!

 

Irrespective of whether or not your family agree with the solution there is a rock solid mathematical proof that it is the case. Agreement or feelings just doesn't come into it. It is proved beyond any doubt, the fact that people don't grasp the logic of the proof doesn't negate the proof.

PS. I hold a mathematics degree and when I first came across the problem I was utterly convinced that the answer could only be 50:50. Until, that is, I saw the proof and thus realised that regardless of my gut feeling, I must be wrong as there was a proof that I was wrong.

 

Play for real money - he'll soon realise his mistake!

 

It's certainly the logical thing to do, since nobody thinks that swopping would reduce the chances of winning the prize.

 

Here's another explanation that might help anyone who is still in doubt:

If you play the game many times over the prize must start off behind door A one-third of the time, door B one-third, and door C one-third.

Since the prize can't move from one door to another, at the end of the game the prize must also be behind door A one-third of the time, door B one-third, and door C one-third.

This means that if you always pick door A and stick with it you can only win one-third of the time. One-third of the time B will be the winner (and the host will open door C); one third of the time C will be the winner (and the host will open door B).

This is why switching doubles your chances of winning from one-third to two-thirds - you win whenever the prize started off behind either B or C.

 

Bob, if you have some examples of how you've used probability in your genealogical research please email them to me - I've got a few of my own, but two heads are always better one.