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Mathematics of Blackjack Insurance Betting Favorable to Players

By Ion Saliu, Founder of Blackjack Mathematics

Favorable blackjack insurance betting based on Ten-cards out, number of players at table.

Figure out the odds to take insurance bet at blackjack and have an advantage, an edge.

Mathematical Foundation of Favorable Insurance in Blackjack

Axiomatic one, I know you are knowledgeable of the game, because you landed here by way of your own research. Of course, you have witnessed many times the blackjack dealer showing Ace as her up-card. The dealer will automatically ask the players: "Insurance?". If desired, any player places half of his initial bet amount in the Insurance box that states clearly a 2 to 1 payout.

The insurance is considered a bad bet because of the house edge it bears: 8% on average. Since I sez average, the house advantage for the insurance bet must go through a series of values, up and down, and down and up... And thusly we reached the point I want to make in this article. There are situations when the insurance bet is favorable to the player. It is mathematical.

In another article I published here, I presented several formulas to calculate blackjack odds, probabilities and the house advantage for insurance. Those formulas and all formulae I present on my site represent undeniable mathematics. Take it to the bank, despite what you've heard from voodoobugs (crooked gambling authors and system vendors). Their saying goes explicitly: "Random events cannot, by definition, be captured in formulae." In reality, the Norman Wattenbergers have no formula to prove the advantage of blackjack card-counting!

No formulas... really??? How about this formula for calculating the house advantage in blackjack insurance betting:

Casino conditions are not favorable to calculating formulas from memory. I'll simplify the calculations to the maximum. The blackjack player will only memorize simple cases when the insurance bet shifts in player's favor. The calculations are based on:

I introduce here the concept of virtual deck. It is impossible to keep track of all the cards remaining in a shoe and apply quick mental calculations. Our concept is founded on a mathematical truth: The cards in a shoe are randomly homogenous. Yes, there are clumps, especially in the first couple of rounds dealt from a manually-shuffled shoe. But everything averages out to a collection of randomly distributed decks.

The player only counts the Tens already dealt to every player in the first two cards. We know the dealer's face card is Ace. The tipping point is one-third (1/3) of Tens. The 10-valued cards amount to approximately one third of total cards in a blackjack deck. The third we calculate here is dependent on the number of players at the table. For example, there are 7 players; they are dealt 14 cards. There is one more card: Dealer's face card. One third is 15/3 = 5. We want the number of Tens already dealt to be below 5; that means, there is a higher proportion of 10-value cards remain in the virtual deck; and thusly a higher likelihood for a dealer's natural 21.

You only have to memorize the figures that show an advantage for the player when taking the insurance. In the case above, with 7 players, the advantage begins at 3 Tens out in 2-card hands. If only 0, 1, 2, or 3 Tens came out, the player has at least a 5.4% advantage when taking insurance. In the extreme situation of NO Tens out in the 14 cards, the advantage for the player reaches 29.7%! Sure, it is alarming for the casinos. They want the blackjack players to take insurance every time they are offered the bet. Instead, the players can easily detect the favorable situations and those are the only times they accept the insurance offer!

Axiomatics, one more factor regarding the virtual deck. We need to see on the table a number of cards close to 13. It has nothing to do with (un)lucky 13 it is related to the number of cards in a suit. 13 also represents a quarter of a deck. The number of players at the table is also important.

The situation changes for 1, 2, or 3 players at the blackjack table. We need to keep the Ten-count for the previous round, plus the current round of 2-card hands. That way we get acceptably close to the 13 count (deck quarter). We can accept also a good average of 3 cards per hand in completed rounds. These three cases are less frequent, as the casinos want all tables full. The increased difficulty is caused by the necessity to always remember the Ten-count of the previous, completed round, and separately the Ten-count of the current round.

Favorable Blackjack Insurance Bet for 7 Players

Favorable Blackjack Insurance Bet for 6 Players

Favorable Blackjack Insurance Bet for 5 Players

Favorable Blackjack Insurance Bet for 4 Players

Favorable Blackjack Insurance Bet for 3 Players

Favorable Blackjack Insurance Bet for 2 Players

Favorable Blackjack Insurance Bet for 1 Player (Heads-Up)

Enhanced Strategy for Favorable Blackjack Insurance Betting

The strategy can become more effective if we work with the half virtual deck, or approximately 26 cards dealt. For that, we need keep accurate counts for one or two previous rounds (completed). I have no doubts that kokodrilos have already thought of such method. (Kokodrilo is royalty name for big-time gambler.)

Everybody can do that. We got the formula and that's the most important tool. Let's do over the workout for the 7-player scenario, the most visible case in any casino. We record the Ten-count of the previous round, with an average of 3 cards per hand, for a total of 24 cards dealt. Add to it the cards we see in the running round of 2-card hands: 15; grand total = 39 (nicely above the half deck).

Let's do over the workout for the 6-player scenario, another frequent situation in any casino. We record the Ten-count of the previous round, with an average of 3 cards per hand, for a total of 21 cards dealt. Add to it the cards we see in the running round of 2-card hands: 13; grand total = 34 (nicely above the half deck).

The 5-player scenario. We record the Ten-count of the previous round, with an average of 3 cards per hand, for a total of 18 cards dealt. Add to it the cards we see in the running round of 2-card hands: 11; grand total = 29 (very close to the half deck line).

The 4-player scenario. We record the Ten-count of the previous round, with an average of 3 cards per hand, for a total of 15 cards dealt. Add to it the cards we see in the running round of 2-card hands: 9; grand total = 24 (very close to the half deck line).

For 3 or fewer players, the difficulty increases as it is necessary to keep separate 10-counts for the previous two completed rounds.

"I received a phone call
From one who claimed was tall.
Then, he cursed me even worse
For my gambling on the horse."

Probability software calculates blackjack natural odds for dealer and several players at BJ table.

Free Blackjack Resources, Basic Strategy, Casino Gambling Systems

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