The dealer shows an Ace. The true count is exactly +3 in both examples. Advantage Count says DECLINE in one and TAKE in the other.
Insurance is a separate wager on dealer blackjack. These two constructed examples compare its expected value (EV) with the Advantage Count v1 engine. Your hand is 10, 8, the offer is still open before the dealer check, and the remaining cards already account for every visible card.
DECLINEInsurance EV −0.0120 units
TAKEInsurance EV +0.0167 units
Dealer shows an Ace · before the check · half-unit Insurance stake
Both states have running count (RC) +12 and four physical decks remaining, giving true count (TC) +3. Their strategy index TC is also +3. The difference comes from the ranks still unseen.
02 / ONE GROUP HIDES A DIFFERENCE
Hi-Lo groups Aces and tens together.
Hi-Lo assigns −1 to A, 10, J, Q and K. Swapping an unseen Ace for an unseen ten-value card leaves the Hi-Lo balance unchanged. With the same number of cards, TC stays unchanged too.
Insurance depends on a more specific event: the dealer’s hidden card has a value of 10. A ten-value card completes blackjack with the visible Ace. Another Ace does not.
| Unseen cards | Example A | Example B |
|---|---|---|
| Aces | 14 | 10 |
| 10, J, Q, K combined | 68 | 72 |
| Aces + ten-value cards | 82 | 82 |
| All unseen cards | 209 | 209 |
A true-count threshold uses a summary of the shoe. The Analyzer divides the remaining ten-value cards by all unseen cards. That extra detail explains why matching true counts can still give different Insurance answers.
The strategy and deviations guide explains this broader distinction between an index rule and a calculation using individual rank counts.
03 / THE TWO POSSIBLE INSURANCE RESULTS
Keep the stake and payout in view.
The application models Insurance as an extra 0.5-unit wager relative to the original one-unit hand wager. A 2:1 winning payout produces +1 unit of net profit; losing costs −0.5 units. Wizard of Odds explains the Insurance wager and payout.
- A: p = 68 ÷ 209 ≈ 32.54%−0.0120 units
- B: p = 72 ÷ 209 ≈ 34.45%+0.0167 units
The break-even probability is exactly one third. Example A falls below it; example B is above it. The engine marks TAKE only when the unrounded Insurance EV is positive. At break-even there is no positive expected gain.
These probabilities come directly from the unseen card counts. Only the displayed values are rounded; there is no simulation sampling interval.
Why divide by 209 when four decks means 208 cards?
The dealer’s hole card is already physically out of the shoe, but its rank is unknown. The Analyzer’s unseen composition includes that card, so it contains 209 possible card identities for this calculation. The physical shoe has 208 drawable cards. Do not subtract an invented hole-card rank.
04 / INSURANCE COMES BEFORE THE CHECK
Read the offer at the right moment.
- Check the visible cards and rules.
For these examples, Your hand is 10, 8 and Dealer shows A. The active rule is Ace peek only. The two starting compositions are listed below.
- Read INSURANCE / DEALER CHECK.
The offer shows Insurance recommendation, EV and P(10-value). Compare all three. P(10-value) concerns the dealer’s hidden card.
- Record the choice, then the actual check result.
Use Take insurance or Decline insurance. Then record No blackjack or Dealer blackjack when that information is known.
If No blackjack has been established, Insurance has already lost when taken. The later main-hand decision is calculated with different information. Do not reuse the earlier Insurance offer as if it were still available.
What changes with European no-hole-card rules?
With European no-hole-card (ENHC), the application uses an Insurance-only step and Continue because there is no early peek. Our two examples use Ace peek only.
05 / ONE DEAL, TWO RESULTS
Insurance settles alongside the main hand.
To make the bookkeeping visible, suppose Insurance is taken. The continuations below are chosen to illustrate settlement. Your hand remains at 18, with Stand recorded on the branches where the dealer check allows play to continue.
| Dealer finishes with | Your hand | Insurance | Combined |
|---|---|---|---|
| A + 10 · blackjack | −1 | +1 | 0 |
| A + 8 · 19 | −1 | −0.5 | −1.5 |
| A + 6 · soft 17, S17 | +1 | −0.5 | +0.5 |
Against dealer 19, both wagers lose. Against dealer soft 17, your hand wins while Insurance loses. Insurance pays for dealer blackjack; it does not cover every way your hand can lose.
The wagers share the dealer’s cards, so their outcomes are related. Still, they have separate payoffs. Keep the Insurance EV distinct from action EV, especially when the latter is shown after a no-blackjack check. Published hand-EV tables also distinguish whether dealer blackjack has been ruled out.
Inspect the starting compositions and rules
Both examples originate from eight decks, or 416 cards. After 207 visible cards, including your 10, 8 and the dealer Ace, 209 ranks remain unknown. One is the reserved hole card, leaving 208 physical cards in the shoe. RC is +12, raw TC +3 and index TC +3.
The common remaining counts are 14 each of 2, 3, 4, 5 and 6, and 19 each of 7, 8 and 9. Example A has 14 Aces and 68 ten-value cards. Example B has 10 Aces and 72 ten-value cards.
The examples use S17, 3:2 blackjack, Ace peek only, all bets lose to dealer blackjack, any-two double, DAS and at most two split hands. Split aces are allowed, without resplit, hit or double of split aces. Surrender and Cash Out are disabled. Penetration is set to 75%; the profile is Extended with floor TC and half-deck estimation.
Insurance uses the supplied composition directly; the strategy profile does not replace that probability calculation. Reproducing only the player hand or TC will not reproduce these states. The original calculations were made with the Advantage Count v1 engine on 13 September 2026.
Sources and further reading
The two compositions and their calculated results are original teaching examples. The sources below support the wager definition and the distinction between decision stages.