Blackjack Odds and Probabiliy - 21 Math and the House Edge
Fri, May 3rd, 2013 | 1:49AM by Charlotte Lee Last Updated Sat, Jan 16th, 2021 | 12:40AM
Online gambling experts talk about the house edge in blackjack all the time. What they discuss is the theoretical online casino advantage in a best case scenario for the gambler. The often discussed payout percentage about 99.5% is based on optimal play. What people don’t always talk about are the probabilities most players face when they don’t understand basic strategy. For a good gambler, the house edge might be 4%, while a bad twenty-one player might face the probability of losing 7% or 8% of all the money they wager.
We want to discuss the rudimentary mathematics of pontoon (AKA blackjack). Once you gain an understanding of blackjack’s basic odds, your decisions during the game will make more sense to you. In this article, we’ll discuss the chances you hit a 21 in various situations and the probability you win while holding any given hand. This in turn gives you a good idea of why you make the moves you do in certain situations. Once you have these concepts learned, some of the basic strategy charts will make more sense and they’ll be easier to memorize.
Chances the Dealer Busts
The reason the house holds an advantage in online blackjack is because you have to act first. It doesn’t matter what the dealer holds. A certain amount of the time, you’re going to lose simply by busting.
The dealer doesn’t need to look at his or her cards to win in this situation. Once the decisions revert to the dealer, then the chances the dealer busts become real.
When a dealer is showing a 2 or 3, the house’s chances of busting range from 35.3% (for a 2) and 37.56% (for a 3).
If the dealer shows a 4 through 6, then that dealer’s hand has between a 40.28% and 42.89% chance of busting.
When the dealer is showing a 7, 8, or 9, their odds of busting are between 23.34% (for a 9) and 25.99% (for a 7).
The ten, jack, queen, and king have a 21.43% of a bust, while the ace only has an 11.65% of busting.
Basic Strategy Versus Dealer Hands
Seeing the dealer’s card tells you a lot about your odds in the hand. If you see an ace, the dealer has a 16% house edge on you. That advantage increases up to 16.9% when you hold a 10, jack, queen, or king, while the dealer has a house edge of 4.3% when holding a 9.
Your best chances of winning come when the dealer is showing a 5 or a 6, because you’re advantage over the casino is over 23%. The next best probability of winning comes when you hold a 4 (18% player advantage), a seven (14.3%), and a three (13.4%). The 2-card and 8-card give you the slightest advantage, but they do hold a significant edge by most gaming standards: 9.8% for a two and 5.4% for an 8.
Two Card Hands
Let’s next look at the odds underlying two-card hands. This is the start to any hand of 21, since players are given two hole cards with which to play. The odds of being dealt a natural 21, called a blackjack, are 4.8%. This is the best hand possible. In the classic game, the worst you can do is tie or “push” when you hold the natural 21. The odds of receiving a hand you certainly stand on (17, 18, 19, and 20) are 30%.
Knowing your odds of busting also lets you play the numbers when it’s time to hit or stand. When you hold a 17, your odds of busting are 69%, or over 2 of every 3 hits. When you hold a 14, the chances of busting are still over 50%, as you have a 56% chance of busting. Those numbers go down quickly with a 13 (39%) and a 12 (30%).
This is a major reason why players who try the “never bust” strategy are losing money. When you hit on a 12 or 13, the odds are in your favour that you’ll improve your hand, so your odds increase in most instances if you hit. Memorize a basic strategy chart before deciding how to play hands. When you do, compare how the advice on the strategy chart is supported by these numbers.
Card Removal Probabilities
The reason card counting works is that the deck composition changes and you’re tracking those changes. Certain cards give you a better chance to win, while others give you worse possibilities.
Removing all twos from a hand increases your odds by 0.4%.
This number increases with the 3, 4, and 5, up to the point where the removal of all 5s from your hand is worth 0.67% to the payout percentage.
On the other hand, removing all 8’s would have a negligible effect, while taking the 10-cards would be worth 0.51% and the ace would be worth 0.59%. So when a number of aces come out of the deck, it is affecting your odds negatively, explaining why you subtract from a card count.
Looking at these numbers also gives insight into why deck size matters. When you’re playing from a 6-deck shoe and an ace comes out of the deck, that’s only 1/24th of that 0.59% or about 0.025% shift in the odds. When you play from a 2-card deck, the impact is proportionally higher (3x).
The shift in your odds would be 0.075%. Not only do you get more penetration in the multi-deck with a lower number of decks, but the effects of a card leaving the shoe aren’t as watered down.
Gambling Games with the Best Odds
Understanding the odds and probability behind the game will help you win more often, lose less money in the long run, and increase your enjoyment of the activity. This page offers just an introduction to these numbers. Some basic study of odds and probability from a math perspective is also a good idea.