Bang or No Bang Game 5: Fastest Finger First

On ITV's "Who Wants to be a Millioniare", in order to get to play for the £1 million prize, you must first compete against nine other contestants in a game called "Fastest Finger First". In this game, you are all presented with four answers which you must put in a certain order. This could be alphabetical, chronological (by date),  in order or height etc.

The four answers are labelled A, B, C and D, so once you figured out the correct answer you have to type it in as quickly as possible. The person who answers correctly the quickest gets to play for £1 million.

But what if you don't have a clue what the correct answer is? Of course you should guess a random order as quickly as possible, eg ADBC, or DCBA or DBCA

How many different ways are there of arranging four letters?

For the first letter you have four to choose from (A, B, C or D). Once you have chosen one, you can't choose it again, so for the second letter you have three to choose from. For the third letter you have two to choose from, and for the last letter you have one to choose from (i.e. no choice at all)

In total then you have 4 x 3 x 2 x 1 = 24 ways of arranging four letters. If you don't believe this, check for yourself. Write down as many combinations of ABCD as you can (making sure that you use each letter only  and exactly once). See; there are 24.

The chance of guessing correctly are therefore

P = 1/24 = 0.042

If you're just guessing, the chances of guessing correctly are really small, even less than guessing the correct section on the Wheel of Fortune

But is there any one combination that is more likely to arise than others? Not if it's totally random. But the producers of "Who Wants to be a Millionaire" very rarely choose "obvious" combinations, like ABCD.

Such a combination is just as likely as any other, but doesn't look like it is random. Because of this human bias in constructing the answers, the ones that look "the most random" occur more offer than ones that don't look random. If was a truly random process no one answer would occur more often than any other.

Which combinations occur most often? DACB and CADB are quite popular! Watch for yourself next time it's on TV.