## Sunday, February 21, 2016

### Blackjack: Approximate Chance of Busting (HP 48G)

Blackjack:  Approximate Chance of Busting

In standard game of Blackjack, the main objective is beat the dealer’s hand.  Usually this occurs if your hand has a better total (no more than 21) than the dealer’s hand.  At any time if the total of your hand exceeds 21, you bust, or automatically lose.

As your score increases, the number of cards that can cause you to bust increase.

If your score is 11 or less, you cannot bust on the next card.  Even if you get an Ace, Aces are 1 or 11, and are always used to your best benefit.

If your score is 12, if the next card is any face card or ten, you bust.  There are 16 face cards and tens in a standard deck of 52 playing cards.  The face cards are Jacks, Queens, and Kings.  In Blackjack, face cards count as 10 points.  Below is a table of score and bust cards.

 Your Score Bust Cards Number of Bust Cards per Deck of 52 Playing Cards 12 Face Cards, Tens 16 13 Face Cards, Tens, Nines 20 14 Face Cards, Tens, Nines, Eights 24 15 Face Cards, Tens, Nines, Eights, Sevens 28 16 Face Cards, Tens, Nines, Eights, Sevens, Sixes 32

If your score is 17 and above, basic Blackjack strategies dictate that you stand (draw no more cards).

Approximating the Odds of Busting

To calculate the odds of busting, we have to determine two things:  (I) the number of bust cards that remain, and (II) the number of cards remaining to be dealt.

The number of bust cards that remain are:

d * (4*t – 32) – b

Where:

d = the number of decks in play.  Decks are assumed to standard decks of 52 playing cards, without jokers.
t = your score.  For our purposes, 12 ≤ t ≤ 16.  Outside this range, this formula does not make sense.
b = the number of bust cards used.  Unless you are counting cards, this is an estimate number.

The number of cards remaining to be dealt:

52*d – n

Where:
d = the number of decks in play (same as above)
n = number of cards that have been played

The odds of busting on the next card is:
OB = (d * (4*t – 32) – b )/( 52*d – n)

Let’s calculate some approximate odds of busting.

Scenario #1:  The table is just you and the dealer.  It is the first hand the session and 6 decks are used.  On the first hand both you and the dealer have been dealt a face card.

Variables: d = 6, b = 2 (two bust cards used), n = 4 cards dealt (one card has an unknown value)

 Score (t) Chance of Busting 12 30.5195 (%) 13 38.3117 (%) 14 46.1039 (%) 15 53.8961 (%) 16 61.6883 (%)

Scenario #2:  Now the table are six players and the dealer.  Five hands have been dealt and now we are on the sixth hand.  There 6 decks being used and let’s assume that 88 cards have been used, 28 bust card used.

Variables:  d = 6, b = 28, n = 88

 Score (t) Chance of Busting 12 30.3571 (%) 13 41.0714 (%) 14 51.7857 (%) 15 62.5000 (%) 16 73.2143 (%)

Scenario #3:  Now the table are six players and the dealer.  Same as above, let’s time assume that 40 bust cards have been used, a lot of high cards have been dealt.

Variables:  d = 6, b = 40, n = 88

 Score (t) Chance of Busting 12 25.0000 (%) 13 35.7143 (%) 14 46.4286 (%) 15 57.1429 (%) 16 67.8571 (%)

A quick observation is that the more bust cards that have been used, the lower the chances of busting later on.  Obviously, having a score of 15 or 16 is not a desirable score since the dealer stands on 17 and above.  15 and 16 are desirable if only a dealer has a weak up card (3-6) and the dealer’s chances of busting is high (assuming the hidden card is a 9, 10, or face card).
The approximations have been calculated using an HP 48GX:

HP 48G Program OBUST:

Stack:  t, d, b, n

≪ → T D B N
≪ 4 T * 32 – D * B – 52 D * N - / →NUM 100 * ≫

And no, I don’t count cards during a blackjack game.  In a real life game, I try to apply basic strategy and hope for the best.

Until next time,

Eddie

This blog is property of Edward Shore. 2016