## Pulsified333 one year ago 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).

1. dan815

|dw:1442546703792:dw|

2. Pulsified333

you are very artistic my friend

3. dan815

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

4. dan815

|dw:1442546916458:dw|

5. dan815

|dw:1442546963874:dw|

6. anonymous

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

7. dan815

|dw:1442547007019:dw|

8. anonymous

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

9. dan815

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

10. Pulsified333

so would it be 4(99/199920)

11. anonymous

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

12. anonymous

.00198

13. Pulsified333

i need a fraction

14. anonymous

5148/2598960

15. anonymous

= 33/ 16660 , oh cool 666

16. Pulsified333

it says that 5148/2598960 is wrong

17. 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

18. Pulsified333

so how is this done?

19. dan815

combinatorics is just logic really

20. dan815

you have to be able to reason through this stuff

21. dan815

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

22. anonymous

33/16660 is wrong?

23. dan815

need to subtract the consecutuve 5 card case from it jay

24. 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

25. anonymous

non straight flushes

26. anonymous

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

27. dan815

close

28. 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

29. dan815

well im not sure they consider KA234 consecutive too actually

30. anonymous

1277/649740

31. anonymous

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

32. 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 )

33. Pulsified333

well I have three attempts remaining

34. dan815

lol

35. dan815

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

36. Pulsified333

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

37. dan815

40

38. Pulsified333

okay thank you

39. dan815

I think you should understand the solution before just answering

40. Pulsified333

I do now

41. Pulsified333

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

42. 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