Fruit Machine Chat

Did anyone catch this on BBC2 last night? He was with an Oxford Mathematics Professor (whose name escapes me now) and they were showing how mathematics can 'change the way you view the universe.

They started off small, however, with an example of how so many people do not understand probability / odds.

He had three cups, one with 'a prize' under and the other two with a booby-prize (find the lady style). They were then shuffled around, and Alan had to try and locate the prize. He didn't, but then was offered the chance to 'change his mind' after it was shown that the first cup he selected had a booby-prize under it. He didn't and picked the remaining booby-prize.

They then expanded the experiment with 20 sets of '3 cups' each and each other selected where they thought the prize would be. However, Alan was not allowed to change his mind after it had been revealed that he had not selected the prize, whereas the Professor could. The results turned out at 15-2 in the Professor's favour...!

Apparently, it is proved that 'changing your mind' in this instance, doubles your chances of winning.

Can any of you math boffins (of which I know there are a few! ) help me out here? You can probably watch it on iPlayer if you didn't see it / my explanation is a bit poor.

it sounds much like this

http://en.wikipedia.org/wiki/Monty_Hall_problem

The film 21 has a good explanation for this too, that's how the main character was inducted into the MIT team as he got the problem spot on in one of his lectures.

This problem was on one of the question sheets for first-year Probability at Cambridge. I remember getting it right, but telling the supervisor that the question was ambiguous because it didn't make it clear that cars are preferable to goats... maybe in Afghanistan or somewhere, goats are more useful.

Istenem wrote:it sounds much like this

http://en.wikipedia.org/wiki/Monty_Hall_problem

Yes, exactly that: a famously unintuitive probability teaser which has tripped many of those that should know better in the past (such as Marilyn vos Savant, the self-styled world's cleverest person). I think the opening post slightly misdescribes it - I'd have thought that after Alan chose, Marcus du Sautoy (assuming it was he) should merely have opened one that Alan hadn't chosen to reveal a booby prize.

Someone once made the point that it becomes intuitively easier to grap if you think of there being twenty separate cups. You pick one of them and the compere opens eighteen of the other nineteen cups, to reveal a lack of prizes underneath each of them. In that circumstance, it "feels" a lot more obvious that you should change i.e. that the probability is right.

It may be easier to appreciate the solution by considering the same problem with 1,000,000 doors instead of just three (vos Savant 1990). In this case there are 999,999 doors with goats behind them and one door with a prize. The player picks a door. The game host then opens 999,998 of the other doors revealing 999,998 goats—imagine the host starting with the first door and going down a line of 1,000,000 doors, opening each one, skipping over only the player's door and one other door. The host then offers the player the chance to switch to the only other unopened door. On average, in 999,999 out of 1,000,000 times the other door will contain the prize, as 999,999 out of 1,000,000 times the player first picked a door with a goat. A rational player should switch. Intuitively speaking, the player should ask how likely is it, that given a million doors, he or she managed to pick the right one.

Stibel et al. (208) propose working memory demand is taxed during the Monty Hall problem and that this forces people to "collapse" their choices into two equally probable options. They report that when increasing the number of options to over 7 choices (7 doors) people tend to switch more often, however most still incorrectly judge the probability of success at 50/50.

This explanation certainly makes it clearer. Sorry Grecian, very much like you say there. Plus, you are also right about my slight mis-representation in the OP...

Interesting stuff...

Yeah, saw that show - was very intertesting. 4 dimentional donuts n stuff!

On a related note - deal or no deal, if the banker offers you a swap, should you?

Hypoteically, if 1p and 250,000 are left... mate insists you should swap as you have a 25 to one shot of picking 250,000 initially so you probably haven't picked it so should swap.

But then the random elimination of the other boxes means surely your odds of NOT eliminating the 250,000 are pretty high. So if its left till the end there is a good chance its in your box... gah. well, 50/50 anyway... annoying.

guess 1p-100,000 are all goats with 250,000 being a car.

More intuitive as the doors/boxes increase.

Not at 1:24, this is cranial dynamite. It's all about that small advantage.
Hmmmmm. I think there's a small advantage in swapping.
Actually that's a guess. Are there three blipples on the side of the box's plastic moulding?

yeah the film 21 does show this it is the 'variable of change' idea that the additional offer provides an extra amount of % in favour of the person making the choice . I am not mathematically minded but I think this is it