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 07-28-2012, 02:58 AM #1 Pooh-Bah   Join Date: Apr 2006 Location: Lafayette, LA Posts: 5,995 Phase 10 odds Not sure if you guys are familiar with this game. The deck contains 108 cards. 4 Skip 8 Wild and 8 cards of each number 1-12. We have six people playing each dealt 10 cards to start. What are the odds of being dealt a ten card straight? It just happened and I think its pretty nuts.
 07-28-2012, 01:44 PM #2 veteran   Join Date: Jun 2011 Posts: 2,263 Re: Phase 10 odds Correct me if I'm wrong, but I'm assuming you got dealt a natural 10 card straight. The math with wilds would be a lot more difficult. For a natural 10-card straight, we have 3 rank sequences possible: 1-10, 2-11, and 3-12. For each rank there are 8 possible cards so this leads to 8^10 = 1,073,741,824 combinations for any given rank sequence. These also include straight flushes. So for the 3 rank sequences possible there are 3*1,073,741,824 = 3,221,225,472 combinations. The number of combinations that exist from the deck of 108 cards: C(108,10) = 108!/98!/10! = 38,722,819,230,810 So the probability of a natural straight (or straight flush) is: 3,221,225,472/38,722,819,230,810 =~ 1 in 12,021 If we want to exclude the straight flushes, we need to subtract 4*2^10 from the 8^10 term since there are 2 cards of each rank in each of the 4 suits. So the number of combinations of a straight only is: 3*(8^10-4*2^10) = 3,221,213,184 Probability: 3,221,213,184/38,722,819,230,810 =~ 1 in 12,021 As we can see, ignoring straight flushes does very little because a 10 card straight flush would be: 4*2^10*3 = 12,288 combos Probability: 12,288/38,722,819,230,810 =~ 1 in 3.15 billion
 07-28-2012, 05:53 PM #3 Pooh-Bah   Join Date: Apr 2006 Location: Lafayette, LA Posts: 5,995 I didn't want to include that for fear of difficulty... but yea it did include a wild card. I had 23456W89T11...it was sick
07-28-2012, 07:14 PM   #4
Carpal \'Tunnel

Join Date: Jun 2005
Location: Psychology Department
Posts: 7,429
Re: Phase 10 odds

Quote:
 Originally Posted by camz2895 I didn't want to include that for fear of difficulty... but yea it did include a wild card. I had 23456W89T11...it was sick
But if it includes a wild card that is going to dramatically change the probability of getting the hand. That means instead of needing 10 unique numbers, you only needed 9 unique numbers and one wild card or the the final unique number. Needless to say there are many more combinations of the latter than the former.

 07-28-2012, 07:33 PM #5 Pooh-Bah   Join Date: Apr 2006 Location: Lafayette, LA Posts: 5,995 Re: Phase 10 odds right...i didnt mean the odds increased...just meant the difficulty of the problem increased
07-28-2012, 07:34 PM   #6
Carpal \'Tunnel

Join Date: Jun 2005
Location: Psychology Department
Posts: 7,429
Re: Phase 10 odds

Quote:
 Originally Posted by camz2895 right...i didnt mean the odds increased...just meant the difficulty of the problem increased
Yeah, I understood what you meant. I was just also mentioning that it does make a big difference in terms of the probability that wilds are included.

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