Chances of getting exactly 21
I ran 1 million simulations on what the chances are getting exactly 21 and how many cards each one took (using standard 8 decks and dealer standing at 17+). these are the results
- 2 cards: 4.7263% (47,263 hands)
- 3 cards: 5.1903% (51,903 hands)
- 4 cards: 1.7872% (17,872 hands)
- 5 cards: 0.3125% (3,125 hands)
- 6 cards: 0.0321% (321 hands)
- 7 cards: 0.0027% (27 hands)
- 8 cards: 0.0001% (1 hand)
this shows that a natural blackjack happened about 4.73% and and multi card blackjacks about 7.33% for a total percentage for the dealer getting a 21 about 12%.
Calculating the chances of the results
To calculate the likelihood of the results you can use the Binomial Probability Formula
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
where P is the probability of success. n is the total number of trials. and K is the number of successes that we are seeing the probability for.
So if we want to find the probability for hitting exactly 27 multi card 21s in 308 trials then you just plug the numbers in. (here is a link to do that https://stattrek.com/online-calculator/binomial)
The chance to get exactly 27 multi card 21s in 308 trials is about 5.13% and the chance to get at least 27 is about 19.21%
The chance to get exactly 13 natural blackjacks in 308 trials is about 8.21% and the chance to get 13 or less is about 26.91%
Chances of the dealer getting a natural 20
A natural 20 is either made up two 10s or an A 9
The chances of getting a natural 20 made of two 10s are (128/416)(127/415) about 7.692%
The chances of getting a natural 20 made of A 9 are (64/416)(32/416) about 1.189%
So the total chances of getting a natural 20 are about 8.881%
The chance of getting exactly 37 (12.01%) natural 20 in 308 trials is about 1.28% and the chance of getting 37 or more natural 20s is about 3.78%
(note: The player also got lucky with natural 20 hands receiving 36 in total with the chance of this about 1.78 and the chance of 36 or more is about 5.56%)
Chances of winning, pushing and losing when a player has a natural 20
I ran another million simulations based for when the player has a natural 20 (consisting of two 10s) and these are the results
- 17: 14.5508%
- 18: 14.0135%
- 19: 13.4172%
- 20: 17.7898%
- 21: 12.0833%
- Bust: 28.1454%
These are the overall results
- Dealer Wins: 12.0833% (dealer gets 21 only)
- Player Wins: 70.1269% (dealer busts or gets 17-19)
- Push (Tie): 17.7898% (dealer also gets 20)
These are the push rates in terms of number of cards drawn
- 2 cards: 10.4049% (dealer natural 20)
- 3 cards: 5.2694%
- 4 cards: 1.7790%
- 5 cards: 0.3025%
- 6 cards: 0.0311%
- 7 cards: 0.0027%
The chance of getting exactly 8 pushes over the 36 natural 20s is about 12.59% and getting at least that many is about 30.38%
Final verdict
Although overall the dealer was quite lucky with some things such as natural 20s and multi card 21s overall. All of the results are really plausible and the overall sample size is way too small to definitively tell whether these results are being rigged.