## Saturday, May 11, 2013

### Win Frequency Overrated

Why does a good blackjack Player stick on bustable hands aginst a Dealer 6?  The quick answer is that with a 6 upcard, the Dealer is likely to bust.  Of course, this is not completely accurate.  The Dealer's bust rate with a 6 is 'only' 42%, which means 58% of the time, he won't bust.  So, first he is not 'likely' to bust.  He is just more likely to bust with a 6 than with any other card.  58% of the time, he will wind up with a 17 through 21 and will beat your hand.  So, why stick?  Well, we need to take into account how often the Player will bust if he takes a hit.  If the Player busts, it doesn't matter what the Dealer does.  This is all a wordy way of saying that the Player is more likely to win if he sticks than if he hits.  Or, in other words, his expected value is higher if sticks than if he hits.  Depending on his specific hand, it might be a relatively small difference between these expected values or it might be a big difference.  But, the difference doesn't matter.  The correct play is the one that has the highest expected value.  This is the key thing to learn for EVERY casino game.

Blackjack is essentially a binary game.  You either win or lose your base wager.  With the exception of blackjacks itself and Doubles and Splits, the wager is a single unit and the outcome is either even money or the Player loses.  Thus, the critical factor becomes win frequency because for the most part, one win is worth as much as any other win.  In video poker, the outcomes are a bit more varied and thus the analysis is actually a good deal more complex.  If we define 'winning' as any hand that is Jacks or Better, that leaves us with a win frequency of 45% (roughly), but not all wins are created equal.  There are essentially 9 different levels of winning, ranging from Royal Flush down to a High Pair.  The payouts range from 800 for 1 down to a push (which is all you get paid when you have a High Pair).

This explains why when playing video poker the win frequency is not very relevant.  Take the following hand as an example:

8♣       9♣       10♣     Q♣      Q♥

There are two ways to play this hand.  A Player can keep the pair of Queens and have a sure winner.  He'll still have a chance to improve to Two Pair, Trips, Full House or Quads.  But, his win frequency will be 100%.  His other choice is to go for the 4-Card Inside Straight Flush.  If he chooses to go this route, his win frequency will be around 30%.  Of the 47 draws, 8 will result in a Flush, 3 in a Straight, 2 in a High Pair and 1 as a Straight Flush.  The other 34 will result in a loss.  If you're motivated by win percentage, then the right play is to stick with the pair of Queens.  If you're motivated to use the proper strategy, you use expected value to guide you.  When the math is all done, we find that the 4-Card Inside Straight Flush has an expected value of 2.39.  The Pair of Queens has an expected value of 1.54.  It's not really much of a choice.  The 4-Card Inside Straight Flush is by far the superior play.

Decisions for casino games are made based on the criteria of expected value.  This is not a concept unique to any particular game.  The same methodology that developed blackjack strategy is essentially the same one used for video poker or Three Card Poker or Ultimate Texas Hold'em.  Some of the toughest decisions are of the type I just described where the Player might have to give up a sure winner to go for a hand that in the long run will pay more, but will have a significantly lower win frequency.  The example I gave here is probably not all that hard to follow.  Since the sure win is only a single unit, it won't feel like you are giving up much.

But, you may have to make a similar decision if you are dealt a Flush that is also a 4-Card Royal.  If you're playing max-coin quarters, you'll be giving up a sure \$7.50 to go for that big payout of \$1000.   IF you're a dollar player, you'll be risking \$30 to win \$4000.  Definitely worth it, but it might just be a little harder to walk away from that sure \$30.