Even though it seems like a dealer always turns a 10 with their second card from this position, the probability that they do complete their blackjack is only roughly 31%. Although this can differ marginally depending on the number of decks in play.
A decision that a player faces when the dealer shows an ace is whether they should accept insurance, which is technically a side bet as to whether the dealer will complete their blackjack or not.
The insurance costs half of the initial bet and is paid at 1/1 if the dealer blackjack materialises. If it doesn’t, this side bet is lost and the original hand plays out as normal.
If an insurance bet ends up being a wise choice, all this effectively does is make the hand a push. A player ends the hand neither in profit or loss.
Had a player bet £20 ahead of the original deal, their insurance would cost an extra £10. The £20 would be a loser against a dealer blackjack, but the £10 would be paid at 1/1, meaning a return of £20.
However, the odds of a dealer not completing their blackjack is 9/4, with four cards in every suit holding a value of 10 and nine worth a different number.
Should the player mentioned above take insurance 130 times, they would win 40 and lose on 90 occasions based on the odds. This translates to £800 in winnings (£20 x 40), but £900 in losses (£10 x 90).
Therefore, the player would be an extra £100 in debt in this period, on top of any losing stakes in the same timeframe.
Many players may wrongly assume that the stronger their hand, the more reason they have to take out insurance.
However, if a player is dealt 20, this is a far weaker hand to insure than that of two low denomination cards. This reason is because of the number of 10s in the deck.
Based on a single deck of 52 cards, there are 16 cards worth 10 and 36 which aren’t. Should a player hold 20 from the deal, consisting of two picture cards, and the dealer show an ace, the remaining 49 cards contain 14 worth 10 and 35 that have a different value.
Dividing 14 into 35 gives an answer of 2.5. So, a player would need to get paid out £25, not £20, for a £10 insurance bet to make it worthwhile. If a player has two low cards, the calculation would be 16 into 33, which equals 2.06.
This shows why a player would be far more likely to win an insurance bet containing two low cards (or any cards not worth 10), compared to a strong hand of 20.
If a player is dealt blackjack against an ace, the situation is slightly different. The decision for the player is whether to guarantee a win of some kind by taking insurance or play for the enhanced blackjack payout of 3/2, which is the best odds typically offered by casinos.
Of the 49 unknown cards on this occasion, there are 15 with a value of 10 and 34 others. If the player is still betting £20 and paying £10 for insurance, they are guaranteed to end the hand with a profit of £20 every time from their initial £20, regardless of whether the dealer has blackjack or not.
If a dealer gets blackjack, a player will receive their initial £20 back for the tie and then £20 for the successful insurance bet. Should the dealer not get blackjack, the player will have laid out £30, but get back £50 as a 3/2 payout on their blackjack.
This means they make £980 overall from these 49 blackjacks.
Had the same player never taken insurance, they will make a £30 profit on 34 occasions and get their initial £20 back on the other 15. This equates to profit of £1,020. Based on these numbers, a player would give away 3.9% of the value of their hand by taking the insurance.
However, there are some rare occasions where taking insurance is the better move.
If playing in a two-deck game of blackjack, there are 104 cards in play. Of these, 32 are either 10s, jacks, queens or kings. When a player is dealt blackjack, the 101 unknown cards consist of 31 10s and 70 non-10s.
Now consider that this player is in the third-base seat on a five-player table and the hand is the first following the dealing shoe being refilled. None of the other four players have a 10 in their hand.
The new situation is that there are 93 unknown cards, containing 31 10s and 62 other cards. This makes it completely evenly balanced whether a player should take the insurance or not.
This is based on an additional eight cards being known (two each from the other four players). Any time nine or more cards are known with no 10s, this enhances the reasoning for taking insurance.
Some blackjack tables can accommodate as many as seven players. In this situation in the final seat, a player could have blackjack with 12 other known cards on the table with no 10s.
Here, there would be 89 unknown cards, consisting of 31 10s and 58 others. Dividing 31 into 58 gives 1.87 and means that a player should expect their £10 insurance bet to pay £18.70 to be completely fair.
The fact they are getting 1/1 and so will be paid £20 for a successful bet, makes it worthwhile.
Similar situations are possible with more than two decks, although it may take a card counter to keep track of lengthy periods without a 10 being dealt.