In n-play, multiple Royals occur more frequently the higher n is. The key to
TomSki question is the "one or more" Royals. Multiple Royals "use up" some of
the expected Royals without adding occurrences to the "one or more" part of
the question.
How about this extreme example. Is it more likely to hit one or more Royals
playing 9000 hands of single line 9/6 JOB, or playing ONE hand of 9000-play? It
is obvious that the AVERAGE number of Royals would be the same, but the fact
that you are certain to hit multiple Royals on the 9000-play (certain if dealt
a RF4, almost certain if dealt a RF3, not to mention the dealt Royal), means
that you can't possibly hit "one or more" as often on the 9000-play. It has
nothing to do with EV. It has everything to do with the way TomSki worded his
puzzle.
Brian
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In a message dated 8/11/2004 8:57:48 AM Pacific Standard Time,
blaw57@yahoo.com writes:
Wait, wait, wait guys, I think you've focusing on
trees and not seeing the forest. I've got the greatest
respect for the math skills of all in this discussion
but you're violating basic assumptions with esoteric
math exercises here. (BTW, TomSki, I just bought your
VPSM program - yes it's still selling - and I love it.
Partly I bought it out of guilt for using a friend's
for a long time. Sure I can calc it myself and usually
do eventually for each game, but with so many games I
start with that. Classic software.)
The odds HAVE to be the same for all options in this
case. Single-line, 3-play and 5-play have the same EV,
so they HAVE to have the same chances of hitting
royals in an equivalent number of hands (hands not
deals). Variance will differ, as the royals will tend
to bunch up in 3- and 5-play, but you can't predict
when that bunching will occur (because it depends on
the deal and you can't predict the timing or order of
when possible deals show up), so the odds of getting a
royal in a given number of hands would have to be the
same.
At least that's my opinion. If I'm wrong, I wish you'd
restart the discussion and help me see why.
Iguana is the only one who laid out math that can be
analyzed. What's wrong with what you did, IMHO, is
that you're measuring the number of deals that will
produce ONE OR MORE royals, but then only counting
that deal as one royal [i.e., 1-(46/47)^5 for 4-royal
draws]. The difference between your results and the
single-line numbers represents the number of multiple
royals from a single deal.
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