The frequency of a dealt 4-to-the-royal is indeed 1 in 2,765. The
expected number of draws for each success is 47. We can look at this
as an "R4 to R" cycle consisting of 47 individual 1 card draw hands,
the same way that we say a quad Deuces cycle in FPDW is 4909 hands
long, or a royal cycle in JOB is 40,391 hands long.
I offer the following for the amusement our board members, to provide
a tool to use in our arsenal of bragging rights, and to decide when
to curse God or praise the Devil ("JUST EXACTLY HOW UNLUCKY I AM???):
"The chance of an rare event not occurring in n cycles is 1 in
(2.71828 to the power n)."
For example, the chance of not getting quad deuces in 15,000 hands
(about a weekend's worth of play) is approximately 1 in
2.71828 * 2.71828 * 2.71828 = 20, since 15,000 hands is about 3 quad
deuces cycles. (n=3)
Example 2 :
A typical year of recreational VP play is about 200,000 hands, which
is about 5 royal cycles. So the chance of a year-long royal dry
spell is about 1 in
2.71828 times itself 5 times, or about 1 in 148. (n=5)
Since there are currently 2,086 members on this board, there are
around 2086/148 = 14 members who are experiencing a (at least) 5
royal cycle long royal dry spell at this moment.
Example 3 :
500 misses in a row in drawing 4-to-the royal? That would be 500/47
= almost 11 cycles, or 1 in 60,000!!! You certainly have something to
brag about. You should have set yourself in front of a 500-play
machine (are those here yet?) and get it over with, 500 blanks all in
one shot! I would love to have this draw for my trophy room!
Notes:
1) The above is applicable to any rare event (1 in 20 is rare
enough) and avoids having to raise a number to a large power like
200,000; 2.71828 is known as the Natural Number (truncated to 5
decimal places); to prove the assertion, simply note that for small
values of p, ln(1-1/p) can be approximated by p, where ln is the
natural logarithm.
2) I credit the late Peter Griffin for bringing this observation
to the gaming public (ch.3, Extra Stuff, 1st ed., Huntington Press,
1991). I also remember reading a related article about two years back
in either Casino Player or Strictly Slots Magazine but can no longer
remember the which issue or the author's name (but I am sure it was
not Henry Tamburin because I don't read Henry Tamburin).
