Sunday, February 09, 1997 2:24 AM
harry026@aol.com

     Norm Michaud reports getting quads five times within about 25 minutes
and wonders about the probability of that happening.  Here is an estimate,
based on the following assumptions:

    1) The flop will be seen if and only if the hand is a pair or two high
cards (T, J, Q, K, or A).  This assumption approximates reasonable play.
    2) After the flop, the hand will be played to the river so long as
there is any chance of obtaining quads.  (Not necessarily reasonable play,
since the turn would be seen holding 22 with the board showing AKQ.  Hope
everyone checked.)
    4) At least one other player stays, allowing you to reach the river.
    3) 12 hands will be played in the 25-minute period.

     Using a Markov chain, these assumptions lead to a probability of a
designated player holding quads by the river in any one hand of
(approximately)  0.0006474603.

     The different hands are identical and independent trials in a
binomial probability distribution.  A "success" will be getting quads, and
we are interested in at least 5 successes among the 12 trials.  Letting
p=0.0006474603, then q=0.9993525397, and summing the expressions

                                (12pickN)*p^N*q^(12-N)

where N goes from 5 through 12 gives (approximately) 0.000,000,000,000,090
(90 quadrillionths) as the probability of getting at least 5 quads among
the 12 hands.  So Norm was quite lucky and should demand very good odds if
he wishes to bet that he can do it again.  However, if the odds are
sufficiently attractive, then he can look upon it as a "jackpot" hand, and
play any two cards he is dealt.  That would increase his starting
probability of getting quads in an individual hand to 0.0016806723 (more
than twice reasonable-play chances).  Then the probability of getting at
least 5 quads becomes 0.000,000,000,010,517; it would happen to Norm on
the average of about once every 100 billion 12-hand groups that he plays.

     As a by-product of the Markov chain are the probabilities of getting
quads at various stages of the hand:

   prob quads by river   at this point in the hand
      0.00064746               before the deal
      0.00842380               pre-flop: pair
      0.00125923               pre-flop: two high cards (TJQKA)
      0.00092507               post-flop: pair
      0.00185014               post-flop: two pair   
      0.04255319               post-flop: set
      0.04347826               post-flop: boat
      1.00000000               post-flop: quads
      0.02173913               post-turn: set
      0.04347826               post-turn: two sets
      0.02173913               post-turn: boat
      1.00000000               post-turn: quads

If *any* two cards are played, then "high cards" can be replaced with "any
cards" and the only number in the above table that would change is at the
top of the column:  0.00168067.

         Harry