A 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. Find the probability that the hand is a Flush (5 nonconsecutive cards each of the same suit).

- Pulsified333

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- dan815

|dw:1442546703792:dw|

- Pulsified333

you are very artistic my friend

- dan815

chance of 5 cards being all diamond + 5 cards being all heart, all clubs , all spades

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## More answers

- dan815

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- dan815

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- anonymous

4 ways to choose a suit * 13 * 12 * 11 * 10 * 9
then divide by 5! because order does not count

- dan815

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- anonymous

( 4 * 13 * 12 * 11 * 10* 9 / 5! ) / (52 choose 5 )

- dan815

the same probability with the other suits so 4*(13*12*11*10*9/(52*51*50*49*48))

- Pulsified333

so would it be 4(99/199920)

- anonymous

( 4 choose 1 * 13 choose 5 ) /( 52 choose 5 ) =

- anonymous

.00198

- Pulsified333

i need a fraction

- anonymous

5148/2598960

- anonymous

= 33/ 16660 ,
oh cool 666

- Pulsified333

it says that 5148/2598960 is wrong

- dan815

you must also subtract the case where the cards are consecutive from this as that would be a straight flush or a royal flush

- Pulsified333

so how is this done?

- dan815

combinatorics is just logic really

- dan815

you have to be able to reason through this stuff

- dan815

lets take one consecutive flush, like A,2,3,4,5
what is the chance this is drawn

- anonymous

33/16660 is wrong?

- dan815

need to subtract the consecutuve 5 card case from it jay

- dan815

okay so this is how i think you should build up the consecutive case
there are 13 pairs of 5 consecutive cards
for example
A,2,3,4,5
2,3,4,5,6
3,4,5,6,7
.
.
.
Q,K,A,2,3
K,A,2,3,4
what is the chance of each of these cases

- anonymous

non straight flushes

- anonymous

( (4 choose 1)*(13 choose 5)-4*10 ) / ( 52 choose 5 )

- dan815

close

- anonymous

there are 10 straights
A,2,3,4,5
2,3,4,5,6
3,4,5,6,7
4,5,6,7,8
5,6,7,8,9
6,7,8,9,10
7,8,9,10, J
8,9,10, J, Q
9,10, J, Q, K
10, J, Q, K, A

- dan815

well im not sure they consider KA234 consecutive too actually

- anonymous

1277/649740

- anonymous

KA234 is a wheel, thats not allowed in texas hold em poker, and most poker variants

- dan815

if I was you id try these 3 solutions for your answer
( (4 choose 1)*(13 choose 5)-4*9 ) / ( 52 choose 5 )
( (4 choose 1)*(13 choose 5)-4*10 ) / ( 52 choose 5 )
( (4 choose 1)*(13 choose 5)-4*13 ) / ( 52 choose 5 )

- Pulsified333

well I have three attempts remaining

- dan815

lol

- dan815

dont make any calculation mistake xD one of those is definately right haha

- Pulsified333

is the 4*10, mean multiply or 4^10?

- dan815

40

- Pulsified333

okay thank you

- dan815

I think you should understand the solution before just answering

- Pulsified333

I do now

- Pulsified333

would the first one's calculation come out to be 5112/2598960

- anonymous

you can use wolfram as a calculator
http://www.wolframalpha.com/input/?i=%28+%284+choose+1%29*%2813+choose+5%29-4*10+%29+%2F+%28+52+choose+5+%29

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