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Expected Returns with Infinite Decks
Introduction
This page shows the player's expected return for any initial hand. It also shows the expected value by just the player's hand, and player's first card.
The tables are based on these rules:
- Dealer does not have a blackjack
- Dealer stands on a soft 17
- Double after split allowed
- Infinite resplitting
- Split aces get one card each
- Infinite number of decks.
Based on these rules the house edge is 0.5093975%.
Expected Value for any Initial Hand
Player's Expected Return Using Basic Strategy
| Player's Hand |
Dealer's Up Card | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Ace | |
| 5 | -0.1282 | -0.0953 | -0.0615 | -0.024 | -0.0012 | -0.1194 | -0.1881 | -0.2666 | -0.3134 | -0.2786 |
| 6 | -0.1408 | -0.1073 | -0.0729 | -0.0349 | -0.013 | -0.1519 | -0.2172 | -0.2926 | -0.3377 | -0.3041 |
| 7 | -0.1092 | -0.0766 | -0.043 | -0.0073 | 0.0292 | -0.0688 | -0.2106 | -0.2854 | -0.3191 | -0.3101 |
| 8 | -0.0218 | 0.008 | 0.0388 | 0.0708 | 0.115 | 0.0822 | -0.0599 | -0.2102 | -0.2494 | -0.197 |
| 9 | 0.0744 | 0.1208 | 0.1819 | 0.2431 | 0.3171 | 0.1719 | 0.0984 | -0.0522 | -0.153 | -0.0657 |
| 10 | 0.3589 | 0.4093 | 0.4609 | 0.5125 | 0.5756 | 0.3924 | 0.2866 | 0.1443 | 0.0253 | 0.0814 |
| 11 | 0.4706 | 0.5178 | 0.566 | 0.6147 | 0.6674 | 0.4629 | 0.3507 | 0.2278 | 0.1797 | 0.143 |
| 12 | -0.2534 | -0.2337 | -0.2111 | -0.1672 | -0.1537 | -0.2128 | -0.2716 | -0.34 | -0.381 | -0.3505 |
| 13 | -0.2928 | -0.2523 | -0.2111 | -0.1672 | -0.1537 | -0.2691 | -0.3236 | -0.3872 | -0.4253 | -0.3969 |
| 14 | -0.2928 | -0.2523 | -0.2111 | -0.1672 | -0.1537 | -0.3213 | -0.3719 | -0.4309 | -0.4663 | -0.44 |
| 15 | -0.2928 | -0.2523 | -0.2111 | -0.1672 | -0.1537 | -0.3698 | -0.4168 | -0.4716 | -0.5044 | -0.48 |
| 16 | -0.2928 | -0.2523 | -0.2111 | -0.1672 | -0.1537 | -0.4148 | -0.4584 | -0.5093 | -0.5398 | -0.5171 |
| 17 | -0.153 | -0.1172 | -0.0806 | -0.0449 | 0.0117 | -0.1068 | -0.382 | -0.4232 | -0.4197 | -0.478 |
| 18 | 0.1217 | 0.1483 | 0.1759 | 0.1996 | 0.2834 | 0.3996 | 0.106 | -0.1832 | -0.1783 | -0.1002 |
| 19 | 0.3863 | 0.4044 | 0.4232 | 0.4395 | 0.496 | 0.616 | 0.5939 | 0.2876 | 0.0631 | 0.2776 |
| 20 | 0.64 | 0.6503 | 0.661 | 0.6704 | 0.704 | 0.7732 | 0.7918 | 0.7584 | 0.5545 | 0.6555 |
| 21 | 0.882 | 0.8853 | 0.8888 | 0.8918 | 0.9028 | 0.9259 | 0.9306 | 0.9392 | 0.9626 | 0.9222 |
| A,2 | 0.0466 | 0.0741 | 0.1025 | 0.1334 | 0.1797 | 0.1224 | 0.0541 | -0.0377 | -0.1049 | -0.0573 |
| A,3 | 0.0224 | 0.0508 | 0.0801 | 0.126 | 0.1797 | 0.0795 | 0.0133 | -0.0752 | -0.1395 | -0.0939 |
| A,4 | -0.0001 | 0.0292 | 0.0593 | 0.126 | 0.1797 | 0.037 | -0.0271 | -0.1122 | -0.1737 | -0.13 |
| A,5 | -0.021 | 0.0091 | 0.0584 | 0.126 | 0.1797 | -0.0049 | -0.0668 | -0.1486 | -0.2074 | -0.1656 |
| A,6 | -0.0005 | 0.0551 | 0.1187 | 0.1824 | 0.2561 | 0.0538 | -0.0729 | -0.1498 | -0.1969 | -0.1796 |
| A,7 | 0.1217 | 0.1776 | 0.237 | 0.2952 | 0.3815 | 0.3996 | 0.106 | -0.1007 | -0.1438 | -0.0929 |
| A,8 | 0.3863 | 0.4044 | 0.4232 | 0.4395 | 0.496 | 0.616 | 0.5939 | 0.2876 | 0.0631 | 0.2776 |
| A,9 | 0.64 | 0.6503 | 0.661 | 0.6704 | 0.704 | 0.7732 | 0.7918 | 0.7584 | 0.5545 | 0.6555 |
| 2,2 | -0.0842 | -0.0153 | 0.0597 | 0.1526 | 0.2282 | 0.0073 | -0.1593 | -0.2407 | -0.2892 | -0.2531 |
| 3,3 | -0.1377 | -0.056 | 0.0305 | 0.1272 | 0.2026 | -0.0525 | -0.2172 | -0.2926 | -0.3377 | -0.3041 |
| 4,4 | -0.0218 | 0.008 | 0.0388 | 0.082 | 0.1516 | 0.0822 | -0.0599 | -0.2102 | -0.2494 | -0.197 |
| 5,5 | 0.3589 | 0.4093 | 0.4609 | 0.5125 | 0.5756 | 0.3924 | 0.2866 | 0.1443 | 0.0253 | 0.0814 |
| 6,6 | -0.2123 | -0.119 | -0.0202 | 0.0824 | 0.1555 | -0.2128 | -0.2716 | -0.34 | -0.381 | -0.3505 |
| 7,7 | -0.1305 | -0.0425 | 0.0508 | 0.1485 | 0.2498 | -0.0485 | -0.3719 | -0.4309 | -0.4663 | -0.44 |
| 8,8 | 0.076 | 0.1485 | 0.2234 | 0.3002 | 0.4127 | 0.3254 | -0.0202 | -0.3865 | -0.4803 | -0.3717 |
| 9,9 | 0.1961 | 0.2592 | 0.3243 | 0.3931 | 0.4725 | 0.3996 | 0.2352 | -0.0774 | -0.1783 | -0.1002 |
| 10,10 | 0.64 | 0.6503 | 0.661 | 0.6704 | 0.704 | 0.7732 | 0.7918 | 0.7584 | 0.5545 | 0.6555 |
| A,A | 0.4706 | 0.5178 | 0.566 | 0.6147 | 0.6674 | 0.4629 | 0.3507 | 0.2278 | 0.1797 | 0.1091 |
The next table shows the expected return for the player's first two cards, before the dealer's up card is known.
Expected Value of First Two Cards
| Player Hand |
Expected Value |
|---|---|
| Hard | |
| 5 | -0.219214 |
| 6 | -0.238748 |
| 7 | -0.213827 |
| 8 | -0.126898 |
| 9 | -0.005401 |
| 10 | 0.205780 |
| 11 | 0.292516 |
| 12 | -0.316036 |
| 13 | -0.347475 |
| 14 | -0.372529 |
| 15 | -0.395793 |
| 16 | -0.417396 |
| 17 | -0.291688 |
| 18 | -0.007066 |
| 19 | 0.265895 |
| Soft | |
| 13 | -0.028239 |
| 14 | -0.055287 |
| 15 | -0.081151 |
| 16 | -0.104935 |
| 17 | -0.078749 |
| 18 | 0.031321 |
| 19 | 0.265895 |
| 20 | 0.579612 |
| 21 | 1.428994 |
| Splits | |
| A,A | 0.277241 |
| 2,2 | -0.146920 |
| 3,3 | -0.189896 |
| 4,4 | -0.123221 |
| 5,5 | 0.205780 |
| 6,6 | -0.246393 |
| 7,7 | -0.247467 |
| 8,8 | -0.145153 |
| 9,9 | 0.066113 |
| 10,10 | 0.579612 |
The next table shows the expected return for the player's first card, before the player's second card or the dealer's up card is known.
Expected Value of First Card
| Player Hand |
Expected Value |
|---|---|
| 2 | -0.134238 |
| 3 | -0.155243 |
| 4 | -0.178283 |
| 5 | -0.197845 |
| 6 | -0.211117 |
| 7 | -0.182054 |
| 8 | -0.087134 |
| 9 | -0.010700 |
| 10 | 0.143489 |
| A | 0.501668 |
Inside Links
- Blackjack main page.
- Appendix 1:Total dependent expected return table for an infinite deck.
- Appendix 2a:Dealer probabilities after dealer peeks for blackjack.
- Appendix 2b:Dealer probabilities before dealer peeks for blackjack.
- Appendix 3a:Composition dependent exceptions to single deck basic strategy where the dealer stands on soft 17.
- Appendix 3b:Composition dependent exceptions to double deck basic strategy where the dealer stands on soft 17.
- Appendix 3c:Composition dependent exceptions to single deck basic strategy where the dealer hits a soft 17.
- Appendix 4:Details on the standard deviation in blackjack.
- Appendix 6:Fine points of when to surrender.
- Appendix 7:Effect of card removal.
- Appendix 8:Analysis of some popular blackjack side bets includingSuper Sevens, Streak, Royal Match, and a tie.
- Appendix 9:Composition dependent expected returns for 1, 2, 4, 5, 6, and 8 decks.
- Appendix 10:The effect on the house edge of the continuous shuffling machines vs. the cut card.
- Appendix 11: Value and strategy for 678 and 777 bonuses.
- Appendix 12:Risk of ruin statistics.
- Appendix 13:Probabilities in the first four cards. May be used to test for the number of decks in online blackjack.
- Appendix 14:Value of each initial player card.
- Appendix 15:House edge using total dependent vs composition dependent basic strategy
- Appendix 16: Basic strategy when dealer exposes both cards.
- Appendix 17: The Ace-Five Count. Possibly the easiest way to count cards.
- Appendix 18: Basic strategy exceptions for three to six cards.
- Appendix 19: Blackjack splitting strategy when a back-player is betting.
- Appendix 20: Blackjack doubling strategy, when doubling after splitting aces is allowed.
- Appendix 21: Details on the Wizard's Simple Strategy.
- "21" Movie — Truth and Fiction : My comments on the movie "21."
- Australian Blackjack: Rules and odds for blackjack down under.
- Introduction to Card Counting
- Rule Variations: The effect of just about every known blackjack rule change.
- Automatic Winner Charlie Rule in Blackjack.
Outside Links
- For a detailed explanation of blackjack rules in Macau, please visit my companion site, Wizard of Macau.
- For a simplified explanation of blackjack, please visit my other companion site, Wizard of Vegas.