Three ideas decide how well you do: counting what's left, knowing when a bet is worth taking, and betting the right amount. Here's each one, with worked examples.
Cards aren't put back, so the chance of higher or lower depends on what's already gone. Count the cards still in the deck on each side of the one showing.
On a fresh deck with a 9 showing, 51 cards are left: 20 higher (10 to ace), 28 lower (2 to 8) and 3 more 9s. Lower wins 28 times in 51, which is 55%.
Now say A♠, A♥, K♣, K♦, K♠, Q♥, 3♣ are gone and a 9♦ is showing. Lots of high cards have gone, so of the 44 left, only 14 are higher and 27 are lower. Lower is now a 61% shot, up from 55%. Same card, much better bet.
Tip: keep a rough tally of high cards (9 to ace) against low cards (2 to 7) as they go. Late in the deck, count exactly, because that's where certainties turn up.
Every bet pays even money: win and you gain your stake, lose and you lose it. So for each £1 you bet, on average you make:
expected profit = p × £1 − (1 − p) × £1 = 2p − 1
Here p is the chance your side wins. Above 50% you expect to make money, and below 50% you expect to lose it. That 2p − 1 is your edge: a 60% shot has a 20% edge.
A card of the same rank counts as a loss for both sides, which is why an 8 on a fresh deck is a skip. Higher and lower each win 24 times in 51 (47%), and the other 3 8s make both sides lose. Neither bet has an edge, so the best bet is none.
Having an edge doesn't mean betting everything. Take a 60% shot and go all in ten times in a row: you only survive if you win all ten, which happens 0.6% of the time. Positive expected value, and you almost always go bust.
The Kelly criterion gives the stake that grows your money fastest over many bets. For an even-money bet it's simple: bet your edge.
Kelly stake = (2p − 1) × your bankroll
In the example above, lower was a 61% shot, so Kelly bets 23% of your bankroll: £227 if you had £1,000. On a fresh deck the same 9 is only worth 10%.
The odds and Kelly stake for every card showing on a fresh deck. After that, what's gone changes everything, so count.
| Showing | Higher | Lower | Best bet | Kelly stake |
|---|---|---|---|---|
| 2 | 94% | 0% | Higher | 88% |
| 3 | 86% | 8% | Higher | 73% |
| 4 | 78% | 16% | Higher | 57% |
| 5 | 71% | 24% | Higher | 41% |
| 6 | 63% | 31% | Higher | 25% |
| 7 | 55% | 39% | Higher | 10% |
| 8 | 47% | 47% | Skip | – |
| 9 | 39% | 55% | Lower | 10% |
| 10 | 31% | 63% | Lower | 25% |
| J | 24% | 71% | Lower | 41% |
| Q | 16% | 78% | Lower | 57% |
| K | 8% | 86% | Lower | 73% |
| A | 0% | 94% | Lower | 88% |
Same-rank cards make up the rest of each row, and they lose either way.
Each bet is compared with Kelly's bet at that moment, using how much of Kelly's expected growth yours kept.
Accuracy is the share of bets rated best or good. Luck decides your final bankroll, but accuracy only depends on your decisions, which makes it the number to improve.
Trading firms use betting games like this to see if you can work out odds quickly, spot an edge, and size a bet without being greedy.