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# Playing it smart - How do you decide between keeping a sure thing and taking a shot?

30 April 2007

How close are you to joy beginning a round of jacks-or-better video poker with four cards to a Royal? The answer depends on the four cards you hold as well as on the one you throw away.

Say you start with 10-J-Q-K of diamonds and something irrelevant such as a six of clubs. You dump the six. Five of the 52 cards in the deck are gone, leaving 47. Only one of these gives you the Royal the ace of diamonds. The chance of pulling that ace is therefore one out of 47. Expressed as a fraction, this is 1/47.

The nine of diamonds gives you a straight flush, the chance also being 1/47. The other seven diamonds (two through eight) yield a flush, so prospects are 7/47. The three off-suit aces or nines make straights. These six cards have a combined probability of 6/47. And you'd recover your bet pairing any of the pictures with any of the nine remaining jacks, queens, or kings. That's 9/47. All told, you have 24 ways out of 47 not to lose altogether. About 51 percent. A bit better than half.

Compare this with an initial J-Q-K-A of diamonds and a useless fifth card. Dump the latter. You again have only one way to obtain a Royal, the 10 of diamonds, so the chance is still 1/47.

There are no non-Royal straight flushes. Any of eight diamonds (two through nine) give you an ordinary flush, with a probability of 8/47. A straight would require one of the three off-suit 10s, at 3/47. And you'd finish with a high pair by pulling one of the 12 remaining face cards 12/47. Overall, that's also 24/47 or 51 percent. From these figures, you can tell intuitively that this starting hand is weaker than the 10-J-Q-K-A, mainly because you've lost the possible straight flush in favor of some pairs.

Consider an inside straight flush with an meaningless fifth card. Make believe it's 10-J-K-A. One card forms the Royal, at 1/47. No ordinary straight flushes. Eight diamonds (two through nine produce a flush) 8/47. Any of the three off-suit queens yields a straight, again 3/47. And nine cards form high pairs for another 9/47. The total is 21/27, somewhat under 45 percent.

These hands are all no-brainers. No self-respecting solid citizen would think twice before tossing the irrelevant cards.

The wicket is stickier when the fifth card makes the initial hand a non-Royal winner. It's then a choice between taking the sure thing, or going for more with the danger of settling for less or even a loss. The high end of this scale would be a suited 9-10-J-Q-K. Hold the nine and you have a certain return of 250 coins for five played. What if you tossed the nine? Your chances are 1/47 of a 1,000-unit return with a Royal, no straight flush, 7/47 of 30 units with a flush (assuming a 9-6 game), 6/47 of 20 units with a straight, and 9/47 of 5 units with a high pair. That's a 23/47 (49 percent) likelihood of any return at all.

You might choose to dump the nine, thinking you'll take the one-out-of-47 shot at the jackpot. Especially knowing you've got almost 50 percent chance of not being totally decimated. And, if the jackpot is the only reason you're gambling, well, don't let anything like the laws of probability stop you.

A more sophisticated player might compare the certainty of 250 coins by holding the nine, with the "expected value" of tossing it. To do so, multiply the chance of each return by the amount, and add the products. In case Fido just chewed up your calcula¬tor, the result is 28.7 units. The certain 250 is much greater so the math tells you to stand pat.

Where do you draw the line? A suited 4-10-J-Q-K has a certain 30 unit return in a 9-6 game. Dumping the four, the chances are 1/47 of the Royal, 1/47 of a straight flush, 6/47 of a flush, 6/47 of a straight, and 9/47 of a high pair. The expected value is 33.4. Now the math says to take the shot. You still may want to grab the sure thing, of course. And, who's to fault you if you do? For, as the beloved bard, Sumner A. Ingmark, reminded us:

The path to success in advance can't be known,
Some people achieve it by numbers alone,
While others reach goals with techniques of their own.