- Eight handed Texas Hold-em game using a 52 card deck.
- This is a 3 1/2 hour session in which there were 105 hands of poker played.
- One Individual achieved the following four poker hands. A. Quad Eights B. Quad Jacks C. Quad Kings D. Club Royal Flush
- 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.
Shouldn't take an expert mathematician.
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.
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.