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:
HA = {(R – T) – T*2} / R
- HA = House Advantage
- R = cards Remaining in the deck
- T = Tens remaining in the deck.
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:
- the number of Tens that came out (you see them on the table)
- the number of cards remaining in the virtual deck.
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 virtual deck represents a 52-card set beginning with the cards we see and counted on the table, or with the previous round.
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.
- For 7 players: The 2-card hands plus Dealer's face-card amount to 15
- For 6 players: The 2-card hands plus Dealer's face-card amount to 13 (right on the nose!)
- For 5 players: The 2-card hands plus Dealer's face-card amount to 11
- For 4 players: The 2-card hands plus Dealer's face-card amount to 9 (lower, but we accept it).
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.
- For 3 players — Previous round (completed): 4 * 3 = 12 cards; current round: 2-card hands plus Dealer's face-card amount to 7; total = 19.
- For 2 players — Previous round (completed): 3 * 3 = 9 cards; current round: 2-card hands plus Dealer's face-card amount to 5; total = 14.
- For 1 player — Previous round (completed): 2 * 3 = 6 cards; current round: 2-card hands plus Dealer's face-card amount to 3; total = 9 (lower, but we accept it).
Favorable Blackjack Insurance Bet for 7 Players
- Number of cards remaining in the virtual deck: 52 - 15 = 37
- House advantage favorable to the player starts at 3 Tens on the table (current round) -
- HA = {(37 – 13) – 13*2} / 37 = (24 - 26) / 37 = -2 / 37 = -5.4%
- The minus sign indicates a house (casino) disadvantage, which means a player's advantage (all players have an advantage).
- The sign turns positive starting at 4 Tens out: +2.7% (advantage to the house, disadvantage to the player).
- Buy insurance if 3 or fewer Tens came out in the running round.
Favorable Blackjack Insurance Bet for 6 Players
- Number of cards remaining in the virtual deck: 52 - 13 = 39
- House advantage favorable to the player starts at 2 Tens on the table (current round) -
- HA = {(39 – 14) – 14*2} / 39 = (25 - 28) / 39 = -3 / 39 = -7.7%
- At 3 Tens out, the result is 0 — no advantage, no disadvantage for the player
- The sign turns positive starting at 4 Tens out: +7.7% (advantage to the house, disadvantage to the player).
- Take insurance if 2 or fewer Tens came out in the running round.
Favorable Blackjack Insurance Bet for 5 Players
- Number of cards remaining in the virtual deck: 52 - 11 = 41
- House advantage favorable to the player starts at 2 Tens on the table (current round) -
- HA = {(41 – 14) – 14*2} / 41 = (27 - 28) / 41 = -1 / 41 = -2.4%
- The sign turns positive starting at 3 Tens out: +4.9% (advantage to the house, disadvantage to the player).
- Buy insurance if 2 or fewer Tens came out in the running round.
Favorable Blackjack Insurance Bet for 4 Players
- Number of cards remaining in the virtual deck: 52 - 9 = 43
- House advantage favorable to the player starts at 1 Tenner on the table (current round) -
- HA = {(43 – 15) – 15*2} / 43 = (28 - 30) / 43 = -2 / 43 = -4.7%
- The sign turns positive starting at 2 Tens out: +4.9% (advantage to the house, disadvantage to the player).
- Buy insurance if 1 or no Tens came out in the running round.
Favorable Blackjack Insurance Bet for 3 Players
- Number of cards remaining in the virtual deck: 52 - 19 = 33
- House advantage favorable to the player starts at 4 Tens out in the current round, plus the previous round -
- HA = {(33 – 12) – 12*2} / 33 = (21 - 24) / 33 = -3 / 33 = -9%
- At 5 Tens out, the result is 0 — no advantage, no disadvantage for the player
- The sign turns positive starting at 6 Tens out: +9% (advantage to the house, disadvantage to the player).
- Take insurance if 4 or fewer Tens came out in the running round, plus the previous round.
Favorable Blackjack Insurance Bet for 2 Players
- Number of cards remaining in the virtual deck: 52 - 14 = 38
- House advantage favorable to the player starts at 3 Tens out in the current round, plus the previous round -
- HA = {(38 – 13) – 13*2} / 38 = (25 - 26) / 38 = -1 / 38 = -2.6%
- The sign turns positive starting at 4 Tens out: +5.3% (advantage to the house, disadvantage to the player).
- Buy insurance if 3 or fewer Tens came out in the running round, plus the previous round.
Favorable Blackjack Insurance Bet for 1 Player (Heads-Up)
- Number of cards remaining in the virtual deck: 52 - 9 = 43
- House advantage favorable to the player starts at 1 Tenner out in the current round, plus the previous round -
- HA = {(43 – 15) – 15*2} / 43 = (28 - 30) / 43 = -2 / 43 = -4.6%
- The sign turns positive starting at 2 Tens out: +2.3% (advantage to the house, disadvantage to the player).
- Take insurance if 1 or no Tens came out in the running round, plus the previous round.
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).
- Number of cards remaining in the virtual deck: 52 - 39 = 13
- House advantage favorable to the player starts at 11 Tenners out in the current round, plus the previous round -
- HA = {(13 – 5) – 5*2} / 13 = (8 - 10) / 13 = -2 / 13 = -15.4%
- The minus sign indicates a house (casino) disadvantage, which means a player's advantage (all players have an advantage).
- The sign turns positive starting at 12 Tens out: +7.7% (advantage to the house, disadvantage to the player).
- Buy insurance if 11 or fewer Tens came out in the current round, plus the previous round.
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).
- Number of cards remaining in the virtual deck: 52 - 34 = 18
- House advantage favorable to the player starts at 9 Tenners out in the current round, plus the previous round -
- HA = {(18 – 7) – 7*2} / 18 = (11 - 14) / 18 = -3 / 18 = -17.8%
- The minus sign indicates a house (casino) disadvantage, which means a player's advantage (all players have an advantage).
- At 10 Tens out, the result is 0 — no advantage, no disadvantage for the player.
- The sign turns positive starting at 11 Tens out: +17.8% (advantage to the house, disadvantage to the player).
- Take insurance if 9 or fewer Tens came out in the current round, plus the previous round.
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).
- Number of cards remaining in the virtual deck: 52 - 29 = 23
- House advantage favorable to the player starts at 8 Tenners out in the current round, plus the previous round -
- HA = {(23 – 8) – 8*2} / 23 = (15 - 16) / 23 = -1 / 23 = -4.3%
- The minus sign indicates a house (casino) disadvantage, which means a player's advantage (all players have an advantage).
- The sign turns positive starting at 11 Tens out: +8.7% (advantage to the house, disadvantage to the player).
- Buy insurance if 8 or fewer Tens came out in the current round, plus the previous round.
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).
- Number of cards remaining in the virtual deck: 52 - 24 = 28
- House advantage favorable to the player starts at 6 Tenners out in the current round, plus the previous round -
- HA = {(28 – 10) – 10*2} / 28 = (18 - 20) / 28 = -2 / 28 = -7.1%
- The minus sign indicates a house (casino) disadvantage, which means a player's advantage (all players have an advantage).
- The sign turns positive starting at 7 Tens out: +3.6% (advantage to the house, disadvantage to the player).
- Buy insurance if 6 or fewer Tens came out in the current round, plus the previous round.
For 3 or fewer players, the difficulty increases as it is necessary to keep separate 10-counts for the previous two completed rounds.
- Caveat: This insurance bet alone can't turn blackjack entirely in the players' favor — the unfavorable sequences are prevalent. Only gambling systems based on streaks can beat the blackjack game consistently.
"I received a phone call
From one who claimed was tall.
Then, he cursed me even worse
For my gambling on the horse."