Rigged or Not: A Basic Statistical Analysis

[Isbjorn]

There are several posts here and in reviews on Google Play alluding to this game being rigged. I thought I would run a very basic test to find out for myself if it were true or not. For background, I hold a PhD in Aerospace Science, and am an experienced blackjack player with many hours (as well as win/loss streaks) in casinos in the US as well as overseas.

For the experiment, I played Stage 2 for 2,007 hands using the basic strategy only. I did not employ card counting. I did not bet hunches, split tens or fives, hit seventeen use any of the other crazy things I have seen some players try in this game. I never bet insurance or took even money for blackjack. Because, there’s suspicion the ‘fix’ is tied to size of the bet, I bet the maximum $5K per hand, doubling and splitting only as prescribed by the basic strategy. Only ~100 hands or so were played on a private table. The rest were on open tables with random players participating.

Normally, in the casino, I use the 1-2-3 back to 1, progressive betting strategy with unit increase by 1 each time my rack grows by 50%. Since I bet the $5K max for Stage 2 this case, I was not able to use this strategy, so I did not track my monetary result. I did track for each of 2,007 consecutive hands (over a few days), dealer up card (2,3,4,5,6,7,8,9,10,J,Q,K,A), dealer final result, my hand final result (win, loss, push, split result), win streak (≥3), loss streak (≥3), dealer bust streak, dealer non-bust streak. The research questions I was interested in were: Was the deal fair? Was the outcome fair? Were there patterns in the deal or outcome which did not track with basic probability or published statistics for blackjack?

Over 2,007 hands, the deal appears to be fair with a few concerns. The dealer’s percentage of Aces, Face cards, and Tens as up card (37.42%) was slightly less than statistically predicted (38.46%). Dealer received slightly (0.63%) more aces than predicted, but less Queens (1.21%) and Jacks (1.31%) predicted. Of the other cards, the only one outside 1% of predicted was 2 with the dealer receiving 1.08% more 2’s than predicted. One other result of note is the dealer received a few more 8’s (0.88%) and a few less 6’s (0.47%) than predicted. These disparities are small and I would think they would flatten out as the sample size increased.

Win/Loss/Push results over 2,007 hands were close to published probability. I won 42.05% of the time versus published probability of 42.22%, lost 49.03% of the time versus published probability of 49.10% and pushed 8.92% of the time versus published probability of 8.48%. I split pairs 2% of the time winning outright 42% of the time, losing outright 23% of the time and pushing 35%. The longest win streak observed was 7 which occurred three times, two more than expected. The longest loss streak was 11 which occurred twice more than expected.

Dealer final hand result gets a little more interesting. The dealer had 13 more blackjacks (0.66%), 18 more 18’s (0.89%), and 10 more 17’s (0.52%) than predicted. The dealer had 16 fewer 21’s (0.78%), and 13 fewer 19’s (0.62%) than predicted. The number of 20’s was within 0.01% of the predicted probability. Of these, the blackjack results are the most concerning. If this were true over a far larger sample, one could easily conclude the deal was rigged. However, I am not sure one can make that case over this sized sample.

The biggest single concern is the dealer bust rate. Expected dealer bust probability for blackjack using a six-deck shoe is ~28.37%. Over 2,007 hands of this analysis the dealer should have busted 569 times. However, this was not even close. The dealer busted 479 times (23.87%), 90 fewer times than predicted probability. This cannot be an outlier.

Drilling down on this is concerning. Of all the possible up cards the dealer can receive, the dealer busts fewer times than predicted probability on all except for 4’s where the bust rate is an almost non-significant 0.65% greater than predicted probability. The dealer bust rate differential ranges from -1.1% for 2’s to -10.12% for 10’s. For all 10 value cards (10, J, Q, K) the dealer bust rate is 7.05% less than predicted. The dealer’s highest bust probability cards, 5 (36.64%), and 6 (36.67%) busted far below the predicted probability rate of 42%. One can conclude the dealer does not bust enough.

Not only does the dealer not bust enough in this game, the dealer does not bust in streaks of hands that defy probability. Over 2,007 hands the dealer had 39 discrete streaks of 10 or greater without a bust. The odds of a single streak of 10 hands without a bust is ~3.5% per ten hands. The dealer had a non-bust streak of 31 consecutive hands and one of 29 consecutive hands (plus a 19 and a 15). The odds of the dealer not busting 31 hands in a row is .2837^31 or 0.00321%. A 31 hand non-busting streak for the dealer (plus the other long streaks) is not normal given this sample size.

The dealer busted three consecutive times 19 times (27 times less than predicted), four consecutive times twice (ten less than predicted), five consecutive times once (three less than predicted), and six consecutive times once exactly as predicted.

So, what’s happening here?

It seems like the initial hand dealt is fair. It would be interesting to see if the percentages I received in this sample would hold up over a much larger sample, but overall, there does not seem to be anything obvious. What is obvious, is the dealer does not bust enough overall and does not bust enough on the cards with the highest bust probability. Both results tilt the game further towards the house and increase the potential of losing more than you should given the same distribution of hands fairly. It would be interesting to analyze what cards the dealer turns over when holding 5 or 6. I felt as if I rarely observed the dealer turn over two high value cards consecutively with these hands, but I did not track this.

One thing which must affect the dealer bust rate is player strategy, particularly splitting tens as many of the players I sat with used this as a regular strategy. Repeatedly doing this over the course of a shoe or a session not only significantly reduces the player’s chance of winning, it also removes potential high value cards from the deck and thus significantly reduces the chance of the dealer busting. While it might be understandable to split tens in pursuit of a game ‘mission’, if one did this in the casino, they would face the full wrath and fury of the other players.

So, is it rigged? It might be. Over 2,007 hands, I observed statistically significant tilt toward the house which hurts the player. Admittedly, this is a rudimentary analysis with a relatively small sample size. Employing the basic strategy in a rigorously disciplined manner plus a progressive betting pattern with a much larger sample would potentially overcome this. The win/loss statistics in this analysis suggest it is possible to achieve the outcomes expected in normal casino blackjack despite the highly abnormal dealer bust rate.