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Best of Alan Krigman

Gaming Guru

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A Million-to-One Odds and that One Player in a Million

26 November 2001

Ever have a casino visit when nothing goes right? When you wonder where you got the idea that gambling was fun? When you figured the law of averages said things had to improve, but they didn't?

My chum, Charlie, was mewling about this very state of affairs a few weeks ago, as I found him meandering aimlessly down the video poker aisles of The Beachcomber. "Jefe," he whined, "they're killin' me. I put $20 into a quarter jacks-or-better machine and ran through the whole thing, five coins at a time, without one hit. The odds against that must be at least a million-to-one."

"Hmmm," I sympathized. "Muy mal, amigo. I don't think I ever bombed entirely that badly. I wonder what the odds really are."

So, naturally, when I got home, I revved up my computer to do the ciphers. Charlie's pretty good at video poker. So his chance of any return at all, based on expert strategy, should be 45.44 percent, give or take. What would be expected if a million players, each this proficient, fed $20 into quarter machines and bet five coins per round? It turns out that 62 out of the million could count on no hits whatsoever. Not quite the odds of a million-to-one that Charlie speculated. But not far off, either.

When the computer had finished chugging away on this calculation, I put the pedal to the metal and went a bit further. A few related questions came to mind. 1) How many of the million players could expect one, two, and so forth up to 16 hits in 16 tries? 2) Recognizing that a high pair only returns what you bet, and is therefore a push, how many of the million players would be forecast to get zero, one, two, et cetera actual wins - two pair or better - in 16 tries? 3) And, what if you raise the bar and ask how many of the million solid citizens would be predicted to have zero up to 16 returns of three-for-one (triplets) or better in 16 rounds? The accompanying table gives the answers.

Number of players out of a million expecting
indicated number of hits in 16 tries
at jacks-or-better video poker

Number of players out of a million expecting
indicated number of hits in 16 tries
at jacks-or-better video poker
hits
jacks
or better
2 pair
or better
triplets
or better
0
62
12,415
153,580
1
822
62,692
305,261
2
5,132
148,399
284,413
3
19,947
218,574
164,882
4
53,990
224,203
66,569
5
107,918
169,829
19,847
6
164,777
98,268
4,520
7
196,049
44,307
802
8
183,688
15,732
112
9
135,985
4,414
12
10
79,278
975
1
11
36,014
168
--
12
12,498
22
--
13
3,203
2
--
14
572
--
--
15
63
--
--
16
3
--
--


As one example of interpreting this data, 62 out of the million players can expect to get zilch in 16 tries, as stated earlier. But 12,415 will experience no actual wins - hands of two pair or better - in this many rounds. And a substantial 153,580 will never see triplets or above in 16 shots.

At the happy end of the scale, three out of a million can expect a return of some kind on every round. Conversely, essentially none ought to hold their breaths for 16 winning hands in a row.

The table also shows intermediate numbers of hits. For instance, 183,688 out of a million players can anticipate any return - from a push on up - exactly eight times in 16 tries. Only 15,732 out of a million are projected to get precisely eight actual wins, as opposed to returns comprising pushes, in 16 tries. And a mere 112 can expect eight hands ranked as triplets or better.

I discussed this with a blackjack buddy, who smugly said "that's what he deserves for playing the machines." So I ran blackjack numbers, too. With six tablemates in an eight-deck game, you'll average about 16 rounds per shoe. Out of a million players, 35 can expect to lose and six to win all 16, respectively. Better than video poker. But hardly sufficient to merit bragging rights. And, one video poker hit could conceivably pay 800-for-1, while blackjack by and large yields only even money. The poet, Sumner A Ingmark, had these wise words about certain fine distinctions:

When a chance is small enough,
Twice the chance seems just as tough.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.