As the story goes, in 1995, what appeared to be a homeless man stumbled into Treasure Island with a $400 Social Security check. By the end of the story, he turned that $400 into a fortune of $1.6 million playing blackjack and then lost it all.

While versions of the story differ on the details, one thing they all emphasize is the man was drunk, rude, treated the employees terribly, and never tipped. It is also said that his blackjack play was extremely aggressive and his strategy was terrible. Doubling on hard 12 or more and splitting tens against anything is something some versions of the story say.

Nobody knows the name of the man. He usually goes by “Shoeless Joe”, the “Shoeless Bandit”, or the “Shoeless Vagrant''. It is alleged that he didn’t wear shoes, even in the bathroom, and that is evidently how he earned these titles. Snopes (https://www.snopes.com/fact-check/the-vegas-vagrant) files the truthfulness of the story as “undetermined.” I personally know a floorman who worked at Treasure Island at the time. He insists that the gist of the story is true.

Snopes is not the only person to wonder about the probability of Shoeless Joe’s amazing run. The Washington Post has an article titled "What are the Odds? Can a Shoeless Guy Walk in and Break the Bank in Las Vegas?" . This article endeavors to answer that question.

The Snopes article says Shoeless Joe played three hands of blackjack at the same time for the maximum of $5,000 per hand. The article does not state the other blackjack rules; but given the time, casino, and stakes, I will assume what I called “liberal strip rules,” as follows:

- Six decks
- Dealer stands on soft 17
- Blackjack pays 3 to 2
- Player may surrender
- Player may double after a split
- Player may re-split any pair, including aces, up to three times

The house edge under these rules is 0.29% with perfect basic strategy. Normally the standard deviation in blackjack is 1.142, but playing three hands at once it is 1.504 per individual hand, or 2.605 for three hands played at once. Playing three hands of $5,000 each has a standard deviation of $7,520 per hand and $13,025 per round. There are many possible outcomes of a hand of blackjack, given all the ways blackjack, surrenders, doubles and splits can play out. To do a pure mathematical solution would have been complicated. I like to do closed form solutions wherever I can, but this problem called for making an exception. So, I modeled Shoeless Joe’s play with a random simulation. Here is how I programmed Joe’s behavior:

- Joe never bet more than half his bankroll, to leave money to double and split, unless he were down to a single $1.
- With less than $12, he played one hand at a time.
- With a bankroll of $12 to $30,000 he bet 1/6 of his bankroll on each of three hands.
- With a bankroll of over $30,000, he bet $5,000 on each of three hands.
- Joe stopped when he went broke or reached a bankroll of $1.6 million.

A perfectionist might argue that Treasure Island wouldn’t accept bets under $5 in blackjack at that time. To that, I would say this is just an estimate and he could have taken his $4 or less to a blackjack machine.

If it were not for the house edge, the probability of success would be 400/1,600,000 = 1 in 4,000.

If it were not for table limits, Joe could have played the pass line in craps, parlaying wins 12 times, to turn $400 into $1,638,400. The probability of success would be (244/495)12 = 1 in 4,859.

However, there were table limits. This would have made it considerably harder to reach $1.6 million, because the more you play, the more the house edge will grind you down. Playing $5,000 a hand, Joe needed to win 320 units of $5,000 each or 160 rounds playing three hands at a time. With the low house edge of these rules, that is still quite doable and probably happens all the time in Vegas.

To prove this, I played out Joe’s scenario 444 million times — each time he played until he went bust or reached $1.6 million. Here are the results:

- Successfully reached $1.6 million = 67,883 times.
- Probability of success = 1 in 6,541.
- Average total amount bet per scenario = $55,186.
- Ratio of total amount lost to total amount bet = 0.29%.
- Success = 3.61 standard deviations above expectations.

With perfect basic strategy, a probability of success of 1 in 6,541 is not remarkable at all. This is comparable to the probability of getting dealt four of a kind in Five Card Stud at 1 in 4,165.

However, whenever this story is told it is emphasized that Joe was a terrible player. To try to model the effect of that, I first introduced a cost of errors of 1%. According to Peter Griffin, this is about the error rate for the average player. To model these errors, I had Joe losing half a bet on an individual hand 1 in 50 hands. I then re-ran the simulation, with these errors thrown in, through 981 million scenarios. Here are the results:

- Successfully reached $1.6 million = 18,304 times.
- Probability of success = 1 in 53,595.
- Average total amount bet per scenario = $28,975.
- Ratio of total amount lost to total amount bet = 1.29%.
- Success = 4.12 standard deviations above expectations.

Notice that the probability of success drops by 88% to 1 in 53,595. This is comparable to hitting a royal flush on the draw in video poker at 1 in 40,388 (based on 9-6 Jacks or Better and optimal strategy).

As mentioned, Joe was said to play worse than the average player. To model that, I increased the cost of errors to 2% and ran the simulation again through 507 million scenarios. Here are the results:

- Successfully reached $1.6 million = 814 times.
- Probability of success = 1 in 622,850.
- Average total amount bet per scenario = $17,529.
- Ratio of total amount lost to total amount bet = 2.28%.
- Success = 4.66 standard deviations above expectations.

With a 2% cost of errors, the probability of success drops by 91% compared to 1%. The probability of success falls to 1 in 622,850. This is comparable to the probability of getting a royal flush in Five Card Stud at 1 in 649,740.

Still, 2% is a bad player, but not horrible. Joe was said to have doubled on hard 12 and would split tens sometimes. What happens if we increase the error rate to 3%? To model that, I ran through my simulator 1.05 billion scenarios. Here are the results:

- Successfully reached $1.6 million = 119 times.
- Probability of success = 1 in 8,823,529.
- Average bet per scenario = $12,260.
- Ratio of total amount lost to total amount bet = 3.29%.
- Success = 5.18 standard deviations above expectations.

With a 3% cost of errors, the probability of success drops by 92% compared to 2%. The probability of success falls to about 1 in 8.8 million. This is comparable to the probability of winning a 6-49 Lottery at 1 in 13,983,816.

Finally, I pushed the cost of errors up to 4%. Here are the results of 1.531 billion scenarios at that error rate:

- Successfully reached $1.6 million = 12 times.
- Probability of success = 1 in 127,583,333.
- Average bet per scenario = $9,398.
- Ratio of total amount lost to total amount bet = 4.29%.
- Success = 5.65 standard deviations above expectations.

With a 4% error rate, the probability of success drops by 93% compared to 3%. The probability of success falls to about 1 in 128 million. This is comparable to the probability of winning the Powerball lottery at 1 in 292,201,338.

My personal opinion on Joe’s error rate is about 3%, based on 35 years of experience playing and studying blackjack. Wherever you put it, between 0% and 4%, the probability of success makes Joe’s story believable. In other words, I would not disbelieve the story, based on the math. On a daily basis, Las Vegas sees people like Joe. These losers leave quietly, and their stories get quickly forgotten. It is the exceptional cases that get remembered.

In conclusion, to answer the question posed by the Washington Post, I put the odds at about 1 in 9 million.

### Comments

I hope he took a break at $1.6 million and bought some shoes. And had a good time doing something before going back and losing it all.