For those who missed it, the full riddle is stated here: https://steemit.com/riddle/@droopy/steem-riddle-of-the-week-11-two-decks
Don't scroll down unless you don't mind spoilers!
Solution
The answer to the problem is quite close to 1-1/e or 0.632.
To solve this you need to calculate the number of "Derangements" which is the subset of the number of permutations of N items where none are in the original order. In combinatorics, often "derangements" of n items is denoted as !n.
Without loss of generality you can assume that one of the decks is in order a,b,c,d,e...
You then know that there are 52! possible orderings for the other deck, of which it turns out, about 1/e are derangements (no element is in its starting position) when n is reasonably large. To learn more about derangements, see here: https://en.wikipedia.org/wiki/Derangement
Prize
This was the first week in several months were no one won the prize!
Don't forget to follow and to check out my past riddles:
https://steemit.com/life/@droopy/steem-riddle-of-the-week-2-flipping-cards-prize-doubled
https://steemit.com/life/@droopy/steem-riddle-of-the-week-3-alternating-truth-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-4-black-cards-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-5-the-cup-game-prize-for-first-correct-answer
https://steemit.com/life/@droopy/steem-riddle-of-the-week-6-burning-ropes-prize-doubled
https://steemit.com/life/@droopy/steem-riddle-of-the-week-7-midpoints
https://steemit.com/life/@droopy/steem-riddle-of-the-week-8-truth-lies-and-chaos
https://steemit.com/riddle/@droopy/steem-riddle-of-the-week-9-wandering-monster
https://steemit.com/riddle/@droopy/steem-riddle-of-the-week-10-weighted-balls