This is the final part of some vaguely mathematical ramblings that started with Deal or No Deal and end up with me proving the obvious by of quantum mechanics.
Part 1 - The Schrödinger Cat Game
Part 2 - To Deal or Not To Deal
Part 3 - You’re reading it now.
After rambling for the last couple of entries I am finally going to get to the point and address the question that started this whole train of thought off, “At the end of a game of Deal or No Deal should you swap you box?”.
This was the question raised at about 6am a couple of weeks back and the short answer is that it doesn’t matter for something like Deal or No Deal which is in essence random. Now the first thing to state is that it doesn’t matter if the placement in the boxes is random or not really, what matters is that the person choosing the boxes one by one is oblivious to their contents. I state this as somebody obviously places the amounts in the boxes and so knows where everything is - Unlike the Schrödinger’s cat example I have used previously, someone knows the contents of the boxes and so in reality there is nothing indeterminate about their states. If I put the £250,000 in box 2 at the start of the game it is there whilst the box is closed and more importantly when the box is opened, it doesn’t just suddenly turn into 1p.
The issue of ignorance is important however, if a decision to remove a box is made and it is made through a position of knowledge and not ignorance it is no longer random and biases the outcome of what is in which box. An example of this is the Monty Hall problem, a problem which shows how bad at probablities people really are.
The difference between Deal or No Deal and the Monty Hall problem is simple and relates to how the contestant goes from three boxes to the final two. In both cases one box is ‘locked’ by the contestant, leaving two boxes to choose between. In Deal or No Deal the contestant chooses a box through ignorance and the contents are revealed before the choice to swap is offered, in the Monty Hall problem however the host reveals the contents of one of the boxes. The difference here is that the host knows the content of the boxes and will never reveal the star prize.
Allow me to demonstrate. There are three identical boxes labelled A, B and C, inside them is one of three prizes randomly distributed for arguments sake the prizes are a car, a goat and a cabbage. (When I tried to explain this originally I made the mistake of using a car and two cabbages, but trying to explain how the order of the cabbages is important not the cabbages themselves is a pain in the arse. At the start of the game there are 6 possible set ups exist:
Box A Box B Box C
Cabbage Car Goat
Cabbage Goat Car
Car Cabbage Goat
Car Goat Cabbage
Goat Cabbage Car
Goat Car Cabbage
The contestant then picks a box at random and has a 1 in 3 chance of picking the door with the prize behind it. (In Deal or No Deal the contestant is assigned a box at the start of the game which for all intents and purposes is picked randomly and arrives at this point after discarding the first 22 boxes and has a 1 in 3 chance of having the highest prize in their box.) In the Monty Hall problem the host then shows the contents of one of the boxes that the contestant didn’t select. Assume for the moment that the contestant choose box A (the actual box isn’t important). Now the host will never reveal the star prize and so his options are limited, if the car isn’t in the box the contestant selected then the host cannot choose which box he eliminates so the above options are reduced to:
Contestant’s Box|
Box A | Box B Box C
Cabbage |Car Goat Has to knock out box C
Cabbage |Goat Car Has to knock out box B
Car |Cabbage Goat Can knock out B or C
Car |Goat Cabbage Can knock out B or C
Goat |Cabbage Car Has to knock out box B
Goat |Car Cabbage Has to knock out box C
Or with the revealed box eliminated it boils down to:
Contestant’s Box|Other Box
|
Cabbage |Car
Cabbage |Car
Car |Goat
Car |Cabbage
Goat |Car
Goat |Car
As you can see here in this scenario the contestant has a 1 in 3 chance of having the car in their box, but if they swap they have a 2 in three chance of winning the car. The application of the hosts knowledge has changed the odds considerably and swapping doubles their chance of winning the star prize if they swap - This runs counter to most people’s intuition, as most people would instinctively think that the odds are unchanged and example of this is my friend to whom I explained this to who even after me developing the above explanation on the spot said that he still wouldn’t swap.
Of course the Monty Hall problem is not the same as the situation a contestant finds themselves with on Deal or No Deal, it is merely similar. In Deal or No Deal the initial combination of 6 boxes applies and one of the boxes cannot be knocked out as it’s the box the contestant has ’selected’. The difference lies in the fact the final box removed is to all intents and purposes random and unlike in the Monty Hall problem the contestant could reveal the contents of the star prize and remove it from the game and so all paths are open:
Contestant’s Box|
Box A |Box B Box C
Cabbage |Car Goat Chooses box B
Cabbage |Car Goat Chooses box C
Cabbage |Goat Car Chooses box B
Cabbage |Goat Car Chooses box C
Car |Cabbage Goat Chooses box B
Car |Cabbage Goat Chooses box C
Car |Goat Cabbage Chooses box B
Car |Goat Cabbage Chooses box C
Goat |Cabbage Car Chooses box B
Goat |Cabbage Car Chooses box C
Goat |Car Cabbage Chooses box B
Goat |Car Cabbage Chooses box C
Which cancels down to:
Contestant’s Box|Other Box
|
Cabbage |Goat
Cabbage |Car
Cabbage |Car
Cabbage |Goat
Car |Goat
Car |Cabbage
Car |Cabbage
Car |Goat
Goat |Car
Goat |Cabbage
Goat |Cabbage
Goat |Car
Now in this case the contestant stills has the same 1 in 3 chance that their box holds the star prize, but the other box also only has a 1 in 3 chance of holding the star prize and so there is no benefit in swapping boxes.
So in conclusion I now not only suspect, but actually know and have proven that if you wind up on Deal or No Deal it is completely random as far as you are concerned and if the banker makes you a decent offer it might just be worth taking it and not fucking it all up by being greedy.
And yes this does mean I have proven the obvious!