dealing three cards to a straight/flush

[yoursmine]

If u deal three cards which is more probable...dealing three cards to a straight or three cards to a flush? I would think a straight would be easier.

For example dealing 8 9 T would qualify for the straight

What are the probabilities?

Please settle this argument

[BruceZ]

[ QUOTE ]
If u deal three cards which is more probable...dealing three cards to a straight or three cards to a flush? I would think a straight would be easier.

For example dealing 8 9 T would qualify for the straight

What are the probabilities?

Please settle this argument

[/ QUOTE ]

If you need 3 consecutive to qualify for the straight, then there are more ways to deal 3 to a flush, but if you just need any 3 cards of a 5-card straight, then there are more ways to deal 3 to a straight.

3 to a flush: 4*C(13,3) = 4*13*12*11/3! = 1144
3 consecutive: 12*4*4*4 = 768
3 out of 5-card straight: (12 + 11*2 + 10*3)*4*4*4 = 4096

For 3 to a flush, there are 4 possible suits times C(13,3) ways to choose 3 cards of each suit.

For 3 consecutive, there are 12 ways to choose the denominations (A23-QKA) times 4 ways to choose the suit of each card.

For 3 out of 5, these can span 3,4, or 5 cards. If they span 3 cards (like 89T), then there are 12 ways to choose the denominations (A23-QKA). If they span 4 cards (like 89J), then there are 11 ways to choose the 2 most widely separated cards (A4-JA) times 2 ways to choose the middle card. If they span 5 cards (like 89Q), then there are 10 ways to choose the 2 most widely separated cards (A5-TA) times 3 ways to choose the middle card. For each of these, there are 4 ways to choose the suit of each card. This includes 3 to a straight flush, but this does not affect which is greater since removing all flushes from the last category would give (12+11*2+10*3)*(4*4*4-4) = 3840.

[yoursmine]

Let me rephrase this one a bit

In the flush example....any three of that suit can come up which I think is what you solved for correct?

In the straight example...the order does not have to come consecutively. For instance you could be given a 7....then a 9....then an 8. The order does not matter

So given that the flush is a little less probable right?
Thanks

[BruceZ]

[ QUOTE ]
Let me rephrase this one a bit

In the flush example....any three of that suit can come up which I think is what you solved for correct?

[/ QUOTE ]

I solved for any 3 of the same suit, e.g. Td 2d Jd, 8s Qs 6s, etc.


[ QUOTE ]
In the straight example...the order does not have to come consecutively. For instance you could be given a 7....then a 9....then an 8. The order does not matter

So given that the flush is a little less probable right?
Thanks

[/ QUOTE ]

I still consider that consecutive. The question is whether it can come 79J since that is 3 cards to the 789TJ straight. If the answer is no, then the flush is a little MORE probable. Otherwise the flush is much less probable.

In 3 card poker, straights are rarer than flushes. This is opposite to 5 or 7 card poker, and this has been mentioned here before.

[yoursmine]

thx Bruce
got it