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March 24

coinflip story doesn't add up

The coinflip story doesn't add up for me: http://www.theatlantic.com/magazine/archive/2012/04/the-man-who-broke-atlantic-city/8900/?single_page=true

Fine, the guy could have a certain confidence that he will make a certain amount, m, since -m gets reduced to -0.8*m by the house discount. But if he keeps on taking his random walk, why would he get from m to 4 million without going back down again and again and again? If it's a cointoss he shouldn't have gotten to 4, 5, 3 million or whatever at all these separate casinos just by playing 100k hands. Something doesn't add up for me. Can you fine researchers investigate the real story here and the math behind it? --80.99.254.208 (talk) 10:54, 24 March 2012 (UTC)[reply]

He was playing blackjack under modified rules. Can you clarify your queston about "cointoss"? RudolfRed (talk) 21:52, 24 March 2012 (UTC)[reply]

The article was far too long and flowery. (TLDR) There's a principle about these kinds of "fluke" events. One million people could have tried to do the same thing and all failed, but they don't write magazine articles about the almost-made-its. Maybe one of those 1,000,000 people manage it. That person isn't special; he's just the one person in a million that manage something that has a one in a million chance of success. — Fly by Night (talk) 23:03, 24 March 2012 (UTC)[reply]

(EC) From the article, the basic situation was this:

The casinos knew that Johnson was loaded and gambled a lot, and they all competed for his custom. This meant making rule changes in his favour, which meant that (assuming Johnson wasn't card counting) the house won 50.25% of the time and he won 49.75%. This seems to be what the OP means by the coin toss.
He was also offered a 20% discount - if he left and cut his losses, he could trim them by 20%. This is the 0.8m the OP refers to.
Perhaps most importantly, he had to float $1,000,000 to play (although apparently he could cut and run at any time, so I don't quite see what this means). Having a huge amount of money reduces the house advantage (casinos hope that a losing streak will wipe someone out - someone with deep pockets can keep playing and potentially win their money back).

So, we can model this as, near enough, a random walk where the forward step (p = 0.4975) is a distance of 100,000 (each bet was $100k), and the backwards step (p = 0.5025) is -80,000. The various rules about splitting and blackjacks and so on make it harder to use a random walk to model this stuff (in other words, his payout should be a little better than 1/0.8, since blackjacks pay more), and I don't know a whole lot about blackjack so I'm mostly working from the figures in the article. It says he played for 12 hours with a peak speed of a hand per minute. Let's say he played 600 hands altogether. A quick mess around on Excel shows that going broke in such a walk, if he's willing to lose $1,000,000, is fairly rare. In these 10 walks, for instance, he only goes broke once, and after 600 hands he's broken even on every single run. He was a bit lucky not to go broke (though from the money he's throwing around, it wouldn't be a massive setback to him even if it did, especially after he won several million in one go), but not all that lucky. Most of those random walks made between $4 and $7 million dollars, after all. A more rigorous test of 240 random walks had him go broke 15 times (that is, 6.25% of the time, or on 1 in 6 games), and his average winnings were $5.8 million dollars (as a sanity check, the expected value is around $5.7 million). It wasn't even that much of a fluke - the biggest stroke of luck seems to be not that he got good blackjack hands, it's that the casino staff he negotiated with were poor mathematicians. If you've got very deep pockets and are good at bargaining, it seems gambling is substantially easier... Smurrayinchester 23:31, 24 March 2012 (UTC)[reply]

(Edit, just realised my average winnings calculation was wrong, since I was including games where he went broke. Fixing that, the average winnings were $5.3 million dollars in 240 games, in which he went broke 20 times). Smurrayinchester 23:54, 24 March 2012 (UTC)[reply]

The casino staff may not be very good at math, but they are certainly good at advertising. Given that he already "broke" a few casinos before, the only reason why the casino would let him on the floor would be for a publicity stunt.Anonymous.translator (talk) 00:12, 25 March 2012 (UTC)[reply]