I've found that the most interesting type of maths is maths which has no practical applications whatsoever. So for no particular reason, I decided to work out the probabilities of some of the more notable boards in DOND history. Without further ado, here they are:

75k Keran - Chances of taking out nine consecutive blues from 11-box: 1 in 55 (I was expecting that probability to be much lower, I must say...)

1p Nick - Chances of taking out all 8 reds consecutively from 12-box: 1 in 495 (this is often overshadowed by him founding the 1p club...)

New Year's Alex - Chances of taking out the five lowest reds in the opening round (or any specific opening round )- 1 in 26,334

75k Daniel - Chances of keeping top four to 4-box: 1 in 7,315

Swappy Matt - Chances of having all-blue first and second rounds: 1 in 1,938 (again, was expecting it to be lower)

Nikki - 5th worst 7-box possible: 5 in 170,544 (or 1 in 34,108.8, but that isn't as neat...)

Brenda - Chances of keeping the top eight to 9-box: 1 in 35,530

Sheran - Chances of having one blue and nine reds at 10-box: 1 in 58,786 (The fact that she had the one blue all along makes it even MORE unlikely, but I couldn't be bothered to work that out. )

Rob/Helen/Otis/anyone else I may have missed: Chances of taking out all of the power 5 in the first two rounds - ~1 in 500 (very ugly number...)

Audrey/Buzz/various other people: Chances of taking out all of the top three in the opening round - 1 in 154

30k Melanie and £50 Alex: Chances of taking out all three penny amounts from 14-box to 11-box - 1 in 308

If there are any other probabilities anyone wants me to work out, then tell me and I'll give it a try. But really, all you really need is a calculator with brackets, and you can do all of the above calculations within 5-10 minutes if you know what numbers to put in. I would highly recommend NOT trying to do it without a calculator though...

Take it you actually enjoyed studying circle theory then?!? (If, of course, you have...)

Loving those statistics

Great probability stats, thanks for posting!

Can't say I've ever found shape mathematics in general that interesting, weirdly enough. Number crunching was always much more intriguing.

Here are a couple more, because I'm really, really bored atm.

Irene/Alan: Chances of taking out four of the power 5 in round one - ~0.88% (roughly 11/1250)

20k Margaret: Chances of taking out nothing over £100 in round one - ~0.2% (4/1881... what an odd fraction )

3rd offer deal Emma: Chances of taking out 6 of the bottom 7 consecutively - 1 in 20,349 (What's even more incredible was that she took out the 250k in the first box!)

So it seems that Sheran's game is the most 'against the odds' game (unless I missed one out). Factor in that she had the one remaining blue at 10-box, and that becomes a 1 in 587,860 chance of doing what she did. To give you an idea of just how unlikely that is, imagine that you've tossed a coin 19 times and it came down heads every single time. Sheran's game was even more unlikely than that!

This begs an interesting question. Considering how games like Brenda's and Sheran's are so amazing in probability terms and almost certainly will never occur again, why have they been more or less forgotten within the show?

The fact you can even remember £30k Melanie let alone her stats makes you an automatic genuis.

Probability of the boxes in Brian's game on 5 March 2010 being opened in the specific order £75k/£35k/50p/£100k/1p/£250/£50/£750/£50k/£15k/£500/£5k/£1k/£10k/£250k/£10/£100/10p/£3k/£20k/£1, with £5 in the player's box: precisely 1 in 1,124,000,727,777,607,680,000.

I doubt we'll see that again anytime soon...

Brilliant.

I just had to be a nerd and work that out for myself, didn't I? Anyway, weirdly (and probably wrongly) I came to the odds of that being:

1 in 1,124,000,727,777,610,000,000 (what do we even call that number anyway?!), based on bringing the £5 to the table being 1 in 22, then picking £75k being 1 in 21... and so on down to £20k being a 1 in 2 pick, and obviously £1 had 100% chance of being picked!

Anyway, it's quite a big number whichever way you look at it!

EDIT: Oh, of course, a sextillion!

Well, fundamental number theory would tell you that the prime factorisation of 22 factorial -- for that is the number we're calculating here -- would contain exactly four 5s. They'd come from the 5, the 10, the 15 and the 20. Those four 5s would combine with four of the 2s in the prime factorisation to make exactly four 10s.

Therefore, 22 factorial must have exactly four zeros at the end, and no more. Which means you're definitely wrong.

(You're right about the sextillion, though. Although old-fashioned people might call it a trilliard.)

In other words, it's very, very unlikely that two contestants will ever open all 22 boxes in the exact same order!

Regarding Brenda's one, as well as keeping all the top eight to 9-box, it was the 4th best possible 9-box. The amount of 9-box scenarios possible is 22!/(13!*9!), which works out to be 497,420, making Brenda's board a 1 in 124,355 chance!




Not really. When Alex did it, Noel claimed that no one had ever done it before, and then everyone got annoyed at Noel for forgetting about Melanie. Though to be fair, when you've been hosting the show for four years and witnessed well over a thousand games, it's a bit much to expect someone in their 60s to remember them all...

Everything of Noel's is in full working order.

Maybe Excel is wrong too then, as it works out 22 factorial to equal the number I said (=FACT(22)), or is that the 'wrong' factorial to be calculating? I'm not a maths teacher, so I wouldn't really know...

You'd probably be better using the scientific calculator that's built in to Windows as I think Excel start rounding numbers between 21! and 22!

A wing related stat:

Chances of having a specfic amount 4 times in a row: 1 in 234,256

Chances of having ANY amount 4 times in a row: 1 in 10,648

This has happened at least three times with contestants on the wings. The most famous example is Michaela having 250k four times in a row, slightly less famously Joanne had 50k four times in a row, and finally Justin who played recently had £10 four times in a row.

Back to in-game stats:

75k Amy - Chances of having 15k or more as the lowest remaining value at 5-box: 1 in 1254

15k Debbie - Chances of having all of the banker's power five at 8-box: 4/1881 or ~0.2% (which happens to be the same as taking out nothing over £100 in the opening round!)

All reds at 11-box: 1 in 705,432. Should happen in about... 2,570 years? Who knows, maybe DOND will still be playing in the year 4576.

More importantly, the chances of me going on the show: 1 in 373248.

Since DOND airs six days a week, all year bar a 4 1/2 week summer break, you would expect that to happen once every 2,475 years (+ a couple of months). So yeah, I wouldn't count on it happening anytime soon...

Also, there's a statistical oddity that you can't work the probability for but is still quite interesting. As of 12th April 2010, 1p has come to the table 56 times, and 29 people have won it. That's over half! Granted, of the 29 people have won 1p, 7 swapped for it rather than having it in their original box, but also 4 people who had the penny swapped it away. So overall, 22 of the 56 people who brought the penny to the table have won it. That must be a higher percentage than any other blue, surely? In particular, this season 7 people have brought the penny to the table in their original box, only TWO didn't win it (Craig and Zoe), and three swapped for it. So there have been more penny winners this season than they have been instances of 1p coming to the table!

Of course, when you look at stuff like Sharron's final offer, Bel's BG of 6.5k on 1p vs 75k and various other occurances, perhaps it's not so surprising with the banker forcing them into their box, essentially (well, except for Corinne, but she was never going to deal with 250k still there anyway...)

Very interesting stats.

What's the chances of having all of the Power 5 at 5-box?

1 in 26334 (the number of combinations of 5 boxes from 22)

What's the chances of taking out all of the Banker's Power 5 in the opening round?

Exactly the same: 1 in 26,334. The chances of taking out a particular combination of five in the opening round or having a particular combination of five at 5-box is 1 in 26,334, because there are 26,334 [22!/(17!*5!)] possible outcomes, but only one of them matches. In fact, here's the total number of combinations it's possible to have at the end of each round:

2-box: 231
17-box/5-box: 26,334
14-box/8-box: 319,770
11-box: 705,432

Considering this, there have been a surprising amount of instances of people having the same 5-box or the same opening round. As far as I know, no two players have yet had the same 8-box or 11-box...