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We asked Bob Dancer to write the answer to today's QoD, which he's always happy to do, being a teacher at heart.
Take it, Bob.
It’s called variance. Assuming you’re playing Full Pay Deuces Wild, four deuces should come around on average every 4,909 hands on average. If you’re playing a different pay schedule, the frequency will be a little different.
If you play 100 “deuce cycles,” meaning 490,900 hands, it will take you about a year if you play 1,000 hands per hour and 20 hours a week. In that time, you’ll get “about” 100 sets of deuces.
Check out the following chart. Column C shows the chances of getting exactly 90 sets of deuces, exactly 91, etc., up to exactly 110. Column D shows the chances of getting exactly those number of deuces or fewer.
In the bold line in the center, you’ll see the chances of getting exactly 100 sets of deuces. It’s a tad less than 4% of the time. It’s more than any other number, but it’s still only one in 25.08 times (shown in column E).
Since “only” 85.289% of the time do you get 110 or fewer sets of deuces, that means you get more than 110 sets almost 15% of the time. You also get fewer than 90 sets of deuces about 17% of the time.
|
A |
B |
C |
D |
E |
|
Number of |
Chance for |
Chance for |
Cumulative |
Inverse |
|
Successes |
1 Hit |
Total Hits |
Chances |
of Total Hits |
|
90 |
0.000204 |
0.025039 |
0.17136 |
39.94 |
|
91 |
0.000204 |
0.027516 |
0.19888 |
36.34 |
|
92 |
0.000204 |
0.029909 |
0.22878 |
33.43 |
|
93 |
0.000204 |
0.032161 |
0.26094 |
31.09 |
|
94 |
0.000204 |
0.034214 |
0.29516 |
29.23 |
|
95 |
0.000204 |
0.036015 |
0.33117 |
27.77 |
|
96 |
0.000204 |
0.037516 |
0.36869 |
26.66 |
|
97 |
0.000204 |
0.038677 |
0.40737 |
25.86 |
|
98 |
0.000204 |
0.039466 |
0.44683 |
25.34 |
|
99 |
0.000204 |
0.039865 |
0.48670 |
25.08 |
|
100 |
0.000204 |
0.039865 |
0.52656 |
25.08 |
|
101 |
0.000204 |
0.039470 |
0.56603 |
25.34 |
|
102 |
0.000204 |
0.038696 |
0.60473 |
25.84 |
|
103 |
0.000204 |
0.037569 |
0.64230 |
26.62 |
|
104 |
0.000204 |
0.036124 |
0.67842 |
27.68 |
|
105 |
0.000204 |
0.034403 |
0.71283 |
29.07 |
|
106 |
0.000204 |
0.032456 |
0.74528 |
30.81 |
|
107 |
0.000204 |
0.030332 |
0.77561 |
32.97 |
|
108 |
0.000204 |
0.028085 |
0.80370 |
35.61 |
|
109 |
0.000204 |
0.025766 |
0.82946 |
38.81 |
|
110 |
0.000204 |
0.023423 |
0.85289 |
42.69 |
So even though you’re “supposed” to get 100 sets of deuces, due to the laws of randomness, the exact number you get can fluctuate quite a bit. If you’re 10 sets of deuces behind, that’s $2,500, assuming you’re playing for quarters. If you’re 10 sets ahead, that’s $2,500 in the other direction -- but no one complains about that.
We could make a similar chart for royal flushes. Over the course of a half-million hands, your net score will be dominated by the number of royals and sets of deuces you can get. And every year it’ll be a different number.
Within that year will be periods of time where you get lots of deuces. I’m sure you’ve had days when you’ve hit four or more sets of deuces and you played nowhere near the number of hands it’s “supposed” to take. Likewise, there are dry spells where you can’t buy a set of deuces. That's what you’re going through now.
When you’re in the middle of a dry spell, there’s no telling when it'll turn around. You might get four deuces on your next hand. You might go another week. You just don’t know. Changing machines or casinos won't change that.
The casino isn’t doing anything nefarious. The machine is just dealing cards. If you’re playing correctly, the long-term average will be about once in 4,910 hands. But over a few weeks or so, anything can happen.
Almost certainly, nothing mysterious happened on December 11 or any other particular date when your machine died. Other players playing the same machines at a different time of day did really well. This is just the way randomness works. If you’ve been gambling 20 years, I’m sure you’ve experienced many dry spells along the way. It’s simply part of the game and part of the nature of randomness.