Chances of a certain set of letters appearing?

[Hannah O]

After a debate with my dad about 5 letter words on Countdown, what are the chances that there will be one round in which the only word that can possibly be made with the given vowels and consonants is IMAGO? Is it likely, unlikely, almost impossible? As my weak point is maths (and I say this often!), can any statistics geniuses explain this in a statistics for dummies way?

Thanks very much!

[Kirk Bevins]

It's impossible for it to be the only word in a selection as AMIGO will always come with it!

[Michael Wallace]

Do you mean the probability that, anagrams aside, IMAGO is the only 5, and there are no longer words?

(edit: Since with things like this it's fairly important to set out the problem precisely)

[Hannah O]

Ah, thanks! I mean precisely what you said, Michael- anagrams aside.

[Charlie Reams]

In 215632 letters rounds on Apterous, it's happened twice (1 2).

[Hannah O]

Thank you very much for your help! Compared to other letters rounds, I'm assuming that the chances of this NOT happening are rather high?

[Charlie Reams]

Thank you very much for your help! Compared to other letters rounds, I'm assuming that the chances of this NOT happening are rather high?

Well, the chance of it not happening appears to be 107815 out of 107816, which is pretty high.

[Hannah O]

Thanks muchly, you have helped me win the debate! Should I lock this topic now the problem is solved?

[Charlie Reams]

We don't usually lock things around here (unless Brian Whitworth starts any more topics.) You might get a more theoretical answer from the Raccoon if you leave it open.

[Michael Wallace]

You might get a more theoretical answer from the Raccoon if you leave it open.

Liek totally.

...

Actually I can't be bothered, because I just started explaining, and it takes long enough to calculate the probability of getting the letters IMAGO in the first place, and that's before you get to the 'difficult' bit. It's theoretically easy, just really really tedious to work it all out.