Probability combination calculation

Please help me out on the card game calculation. Similar with 3 card poker but with 8 decks of playing cards and It total combination is combin(416,3)=11,912,160. So what would be the combinations, probability or I in how many hands of Royal Flush, Straight Flush, Flush, Straight, three of a Kind, One pair and Nothing in a shoe with 8 decks of playing cards?

Quote: SV

Please help me out on the card game calculation. Similar with 3 card poker but with 8 decks of playing cards and It total combination is combin(416,3)=11,912,160. So what would be the combinations, probability or I in how many hands of Royal Flush, Straight Flush, Flush, Straight, three of a Kind, One pair and Nothing in a shoe with 8 decks of playing cards?

Royal flush- 3/13 x 16/415 x 8/414 = or 1 in 97,763

Flush- 103/415 x 102/414 = or 1 in 16.35 (some of these will be Royals, or staright flushes, or even 3 of a kinds)

3 of a kind- 31/416 x 30/415 = or 1 in 185

My brain doesnt want to do the rest...

The problem would be very similar to calculating the odds/probabilities for a three card bonus in a 21 plus 3 game. Give me a couple days to work on it and I'll get it posted. My initial guess would be to figure it for one deck then multiply it out by 8

Here are the results of a very quick (and therefore possibly incorrect) calculation:

Mini Royal = 2,048
Straight Flush = 22,528
Three of a Kind = 64,480
Straight = 368,640
Flush = 703,840
One Pair = 2,476,032
High Card = 8,274,592
Total = 11,912,160

Quote: johnsm2010

My initial guess would be to figure it for one deck then multiply it out by 8

This can't be right. All probabilities must sum up to 1 - they don't sum up to 8 just because you use 8 decks.

I would start from an infinite deck calculation - the differences to 8 decks are marginal.
Then if it is worth the effort, make the small step to 8 decks by adjusting to the finite size
(i.e. multiply with 416/416 * 416/415 * 416/414 for each hand containing 3 distinctive cards, ...)

Quote: JB

Here are the results of a very quick (and therefore possibly incorrect) calculation:

Mini Royal = 2,048
Straight Flush = 22,528
Three of a Kind = 64,480
Straight = 368,640
Flush = 703,840
One Pair = 2,476,032
High Card = 8,274,592
Total = 11,912,160

I get 700,928 flushes (2,912 less than you, which is the number of suited trips)
I get 2,336,256 pairs (139,776 less than you, which is also the number of pairs that are also flush)
I get 8,417,280 high cards (142,688 more than you 2,912 + 139,776)

Quote: miplet

I get 700,928 flushes (2,912 less than you, which is the number of suited trips)
I get 2,336,256 pairs (139,776 less than you, which is also the number of pairs that are also flush)
I get 8,417,280 high cards (142,688 more than you 2,912 + 139,776)

Agreed. I neglected to take suited pairs & trips into account.