probability of getting flush with suited hand

[AaronBrown]

ChicagoVince's method works pretty well, here is the exact way to do it. You have to figure the cases of getting exactly three, four and five suited cards separately.

Three suited cards: C(11,3)*C(39,2) = 122,265
Four suited cards: C(11,4)*C(39,1) = 12,870
Five suited cards: C(11,5)*C(39,0)= 462

To get the probabilities, you have to divide by C(50,5) = 2,118,760. You get 5.77%, 0.61% and 0.02% respectively, for a total probability of 6.40%.

[TheProdigy]

Hey,

I'm just wondering, when you use C(39,2), C(39,1), etc on your three examples, what is the reasons for using those numbers?

Still trying to learn Probabilities,
Prodigy

[BruceZ]

[ QUOTE ]
Hey,

I'm just wondering, when you use C(39,2), C(39,1), etc on your three examples, what is the reasons for using those numbers?

[/ QUOTE ]

There are 39 non-flush cards and 11 flush cards remaining. C(39,2) is the number of combinations of 2 non-flush cards, and C(11,3) is the number of combinations of 3 flush cards, so C(11,3)*C(39,2) is the number of ways to make a flush with exactly 3 flush cards and 2 non-flush cards on the board.

C(39,1) = 39 is the number of non-flush cards, and C(11,4) is the number of combinations of 4 flush cards, so C(11,4)*C(39,1) is the number of ways to make a flush with exactly 4 flush cards and 1 non-flush card on the board.

[Mike Haven]

For anyone who can't remember why, for example,
C(11,3)*C(39,2) should equal 122,265 this might help:

C(n,r) is often written as nCr

and it equals (n!)/[(n-r)!r!]

(note: n! is spoken as "n factorial" and means multiplying all the numbers from 1 up to n together; so if n = 4, n factorial would mean 1*2*3*4 or 1x2x3x4 or 4x3x2x1 or 24)

which for 11C3 means (11x10x9x8x7x6x5x4x3x2x1)/[(11-3)!3!]

or (11x10x9x8x7x6x5x4x3x2x1)/(8x7x6x5x4x3x2x1)x(3x2x1)

(note: you can simply cross off the (8x7x6x5x4x3x2x1) from the end of the (11x10x9x8x7x6x5x4x3x2x1) without having to multiply them both out, which will give

or (11x10x9)/(3x2x1)

or 165

and for 39C2 means (39x38x...)/[(39-2)!2!]

or (39x38x...)/[(37x36x...)x(2x1)]

or (39x38)/(2x1)

or 741

so 11C3*39C2

or C(11,3)*C(39,2)

or C(11,3) x C(39,2)

or 165 x 741

equals 122,265.

I hope is useful to someone, and if you save these calculations they might make the next combinations thread a little easier to understand.