Recently I was asking myself questions like, "what are the odds of
hitting a Royal Flush in any given hour of play? How many hands do I
have to play, on average, to get one?"
This is going to depend, of course, on the specific game that you're
playing. Each game has its own "Royal Flush Cycle." For the
familiar 9/6 Jacks or Better, the RF Cycle is 40,390. For most other
games, it's somewhat more. But what exactly does this mean? If I
play 40,389 hands without a RF, do I say "Hot dog, I'm gonna hit
it next hand!"? Of course not no matter what has happened in
previous hands, the odds for the next hand are the same as any other. Is
that the average number of hands I need to play to get the RF?
Apparently, it's not.
The R.F. Cycle means simply this: that is the odds against hitting the
RF on any given hand. For each hand I play of 9/6 JoB (assuming proper
playing strategy, of course), the odds are 1 in 40,390 of getting the
RF. As a decimal, that number is .000024759; let's call that p. More
interesting perhaps, and surely more useful, is 1-p, the odds that I
will not get a RF. That probability is .99997524. For each subsequent
hand, we multiply that number times itself to get the odds of no RF. The
odds of not getting the RF in n hands is (1-p) raised to the power n.
When I play 1000 hands of JoB (roughly 2 hours' play), the odds are
only 2.445% that I will get at least one RF. Now let's play some
more hands, and see how that number increases:
5,000 hands = 11.644% of getting the RF
10,000 hands = 21.93%
15,000 hands = 31.02%
20,000 hands = 39.05%
25,000 hands = 46.15%
30,000 hands = 52.42%
35,000 hands = 57.96%
40,000 hands = 62.86%
45,000 hands = 67.18%
50,000 hands = 71.00%
60,000 hands = 77.36%
70,000 hands = 82.32%
80,000 hands = 86.20%
90,000 hands = 89.22%
100,000 hands = 91.59%
125,000 hands = 95.47%
150,000 hands = 97.56%
200,000 hands = 99.29%
So we see that it is entirely possible to play 100,000 hands or more
without getting the RF, although it is unlikely.
A very interesting number is 28,000 hands, because the probability is
almost exactly one-half that you will see the RF: 50.005%. Play another
28,000 hands (56,000 total), and the odds of not seeing the RF drop from
1 in 2 to 1 in 4. Play 28,000 more (84,000), and the odds of not seeing
it drop to 1 in 8. And it continues to fall in half each time you play
another 28,000 hands but it never goes to zero!!
So hopefully this little exercise will help players to "set their
expectations" about their chances of getting the RF in any given
number of hands, or hours of play.
Robert
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