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Best of Alan Krigman
Are You Overestimating Your Chances of a Hit?14 January 2003
There are no such screws, of course. The problem, common to most forms of gambling, is that few solid citizens have much of a grasp on the likelihoods of the outcomes they either desire or dread. Completing a four-card straight, flush, or straight flush ought to be a piece of cake. Right? Well, let's see.
Take a double-ended outside straight such as 4-5-6-7, mixed suits, nothing wild. You've seen five cards -- four you've held, one you tossed. So 47 remain in the 52-card virtual deck. To win, you need any of eight cards, four threes or four eights. Your outlook is to average eight hits in 47 tries, only 17 percent.
Inside and single-ended outside straights, for instance 4-6-7-8 and
A-2-3-4, respectively, are even tougher to complete. With nothing wild,
only one rank -- four possibilities -- will do the trick: any five in
either of these cases. The prognosis for success is four out of 47, a
mere 8.5 percent.
Perhaps you prefer joker-poker. Now, 4-5-6-7 means you need the joker or any three or eight -- nine out of the 48 remaining, or close to 19 percent. Easier, but not by much. If one of the cards you hold is the joker, your prospects depend on where it is. On an end is best. For example 5-6-7-X completes with any three, four, eight, or nine. This is 16 out of 48 or 33 percent. In the middle, maybe 5-X-7-8, you'd win with any four, six, or nine: 12 out of 48 or 25 percent. A joker in the hand also beats one in the deck because matching any hard rank already held yields triplets; that's nine cards out of 48, around 19 percent chance.
How about flushes? When nothing's wild, starting with four of the same suit means nine are left in the 47 remaining cards. Your chance is nine out of 47, just over 19 percent. With one joker wild, this goes up to 10 out of 48 -- roughly 21 percent. One of the four cards you hold already being the joker doesn't change the probability of a flush. You've used three of the target suit so 10 are left in the 48 remaining cards. But, as with a joker and three to a straight, you can make triplets if the flush falls through, with the chance also being nine out of 48 or 19 percent.
Possible straight flushes offer more possibilities. Say your game is nothing wild. Your top shot would be an outside four-card possible straight flush. Considering high pairs, best is a suited 10-J-Q-K. You have two ways out of 47, 4.25 percent, to make the straight flush. This leaves seven ways out of 47 (almost 15 percent) for a flush, six out of 47 (nearly 13 percent) for a straight, and the same 19 percent for a high pair. That's close to 51 percent in all. An inside or single-ended straight flush drops you to one card out of 47 for the straight flush and three out of 47 for the straight, 2.1 and 6.4 percent, respectively. The chance of a flush is again nine out of 47, or 19 percent.
With a joker, the hardiest four-card possible straight flush is the 5-6-7-X type. Four cards make a biggie (8.3 percent), 10 a flush (20.8 percent), 12 a straight (25 percent), and nine trips (18.8 percent) -- 72.9 percent prospects of at least some return.
So it's in probabilities, which aren't all that high, not in how you talk to slot attendants. But, be gracious to those who serve you in the casino anyway. As the poet, Sumner A Ingmark, advised:
Guard against antisocial proclivity,
Best of Alan Krigman