# This is a probability question which will require an expert mathematician's expertise. Below is the criteria

1. Eight handed Texas Hold-em game using a 52 card deck.
2. This is a 3 1/2 hour session in which there were 105 hands of poker played.
3. One Individual achieved the following four poker hands. A. Quad Eights B. Quad Jacks C. Quad Kings D. Club Royal Flush
4. Caveat - This man is blind in one eye, has one arm and was wearing a short-sleeved shirt - no chance of cheating. I know this sounds like a ridiculous fallacy, but I was playing and witnessed this amazing occurrence. I have given this problem to a few, expert mathematician friends to calculate the odds and am interested to see their answer. Please include equations/formulas and process used.
• Not exactly an answer to your question, but if this happened to me I would be very suspicious of cheating, especially if the cards are being handled by the players and not a dealer in a casino. There are many ways of cheating besides an ace up your sleeve, for example the legally blind card mechanic Richard Turner can do some amazing things. Sep 16, 2022 at 18:18
• The problem is that whilst this exact scenario is incredibly unlikely, there are many other very similar scenarios (e.g. two straight flushes, or four quads, or six full houses). Any one of those scenarios would have led to your asking a similar question. The chance of one of those scenarios occurring is (whilst still unlikely!) much less unlikely. So it's hard to give a meaningful answer without being more precise about what probability you're actually asking for. Apr 1 at 10:45

Shouldn't take an expert mathematician.

See image: Quad Odds = .2% Royal Odds = .00015%

Total Odds of Hitting Either are .20015% in any hand. Total Odds of Hitting Either is 21.015% over 105 hands Same calculation for 104,103,and 102 hands and you come up with a cumulative .1841% chance that over 105 hands you will hit any combination of 4 quads or Royals.

Exactly 3 Quads and 1 Royal multiplies the 21%,20.8%, and 20.6% from the quads by the .02% from royal and you get .0001417% chance that this exact run would occur.

Another way of stating would be that you would need to play 70,561,791 x 105 hands or 7.41 Billion hands to achieve this outcome within any set of 105 hands.

Lmk what you think.

• Also, I hope this guy took full advantage. Would be interested to hear how much he was up ;) Feb 16, 2022 at 19:30
• How do you go from 0.000141% to 7+ billion hands? 1/0.000001417 is about 700k, not 70 million Apr 12, 2022 at 12:47
• This maths is way off beam. First, you appear to have taken the royal flush odds (0.00015%) from the 'Poker Probability' page on Wikipedia. This is the chance of making a royal flush with five random cards, not two hole cards and the board. Your odds for quads are something different again. And the rest of the calculations also make no sense. Apr 1 at 10:37

The probability of getting quads on a particular had is 1/407. For a royal flush, it's 1/30,940. The probability of getting three quads and one flush in a particular order (e.g. first three hands are quads, fourth is a flush) is (1/30,940)(1/407)^3*(1-1/407-1/30,940)^101. There are 105*(104 choose 3) ways to choose the order. If we take (1/30,940)(1/407)^3*(1-1/407-1/30,940)^101*105*(104 choose 3), we get 0.00000712652, which is about 1/14,0321.

If you literally want the probability of those exact quads, and specifically a club royal flush, then it's (1/30,940)(1/(13*407))^3*(1-1/(13*407)-1/30,940)^101*(105 permute 4) = 2.44803379e-8 or about 1/40,849,109.

The first number is the probability for getting exactly three quads and exactly one royal flush. The probability of getting at least three quads and at least one royal flush is slightly higher. The second number is for getting exactly one quad 8, one quad jacks, and one quad kings, ignoring all other quads.