We're going to jump all the way back to 2009, to the 60th episode of Scam School (At this writing, that was 183 episodes ago).
In that episode, host Brian Brushwood presented a scam whose odds sounded almost too good to be true. I'll investigate the actual odds in this post.
In episode 60 of Scam School, there are 2 scams that are taught. This post focuses on the Playing The Odds scam which starts at the 4:33 mark in the following video:
What are the chances of two named values being together in a deck of cards? Brian mentions his experience of the probabilities in his write-up:
Amazingly (and to just about everyone's disbelief), it seems that about 70% of the time, any two named values will just happen to be side by side in a shuffled deck of cards!It's not easy to develop probability equations for this challenge. Just defining all the possible arrangements involved is a challenges. I don't doubt that this is why Brian gave up playing with the numbers, and turned to brute force calculations, otherwise known as the Monte Carlo method.
(by the way, math wizards: if you can figure out a way to calculate the exact odds on this, I'm all ears. After hours of playing with the numbers, I finally gave up and just did a brute force calculation: after 50 trials, I ended up averaging about a 70% success rate)
James Grime filmed a response video in which he explains the difficulty of calculating the odds via equations, and the result of his own Monte Carlo simulations:
The video shows a probability of 48.3%, and the information box in the video says that other experiments moved that closer to 48.6%.
After watching this video, I wrote and ran my own Monte Carlo simulations in jQuery. I had the computer mix the deck using this implementation of the modern Fisher–Yates shuffling algorithm, which a quick pencil-and-paper exercise will make clear.
After running 10 million trials of my own simulation, my results suggested a 48.63627% chance of succeeding, effectively the same 48.6% chance described above. In short, the person betting against the 2 values showing up next to each other will win roughly 51.4% of the time. With such a low probability of success, how did this bet manage to become popular?
The first thought I had about this was that perhaps it involved paying less than true odds. The odds of you winning this bet are roughly 1.056 to 1 against. In other words, as long as you can convince someone to bet at least $1.06 to every $1 you bet, you could still make money with this bet over the long term. That doesn't seem very likely.
Many bets hinge on a little wordplay. For example, there's a classic bet where you claim you can name the day someone was born, with an accuracy of plus or minus 3 days. Once they put up their money, you simply say Wednesday, and take their money. Since every day of the week is plus or minus 3 days from Wednesday, you can't lose.
In a similar manner, perhaps we can use wordplay to give us a better margin of error for this bet. What if, instead of mentioning that the cards must be next to each other, the bet was that the two values would be within 1 card of each other? If the two cards show up right next to each other, as in the original bet, this sounds exactly like what you bet. In addition, it also covers the possibility of the 2 values showing up with 1 card between them.
I re-programmed my simulation to include the new possibilities, ran it another 10 million times, and came up with about a 73.6% chance of success, or odds of roughly 2.8 to 1 in favor of winning!
Brian's own test trials intrigue me. Assuming that he wound up winning 34 out of those 50 times, which seems reasonable given the about 70% phrasing, Wolfram|Alpha says there's only about a 0.44% chance, or odds of about 224 to 1 against winning 34 or more out of 50 such trials! As with any trials, though, long shots can and do happen.
Alternatively, that claim might be a scam...