Probability of hitting a flush from flop to turn

[Maverick511]

I know in holdem if you have four to the flush on the flop, you have a 19% chance to hit your flush on the turn and a 19.5% chance to hit it on the river. However if you are all in on the flop, you have a combined 35% chance to hit it from flop to river. Can anyone show me the math used to calculate the 35%??


Thanks
-=Maverick=-

[BruceZ]

[ QUOTE ]
I know in holdem if you have four to the flush on the flop, you have a 19% chance to hit your flush on the turn and a 19.5% chance to hit it on the river. However if you are all in on the flop, you have a combined 35% chance to hit it from flop to river. Can anyone show me the math used to calculate the 35%??


Thanks
-=Maverick=-

[/ QUOTE ]

You already asked that, and got perfectly good answers (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Board=probability&Number=106 3497&Forum=,All_Forums,&Words=&Searchpage=0&Limit= 25&Main=1063497&Search=true&where=&Name=6964&dater ange=&newerval=&newertype=&olderval=&oldertype=&bo dyprev=#Post1063497) , so why ask again?

[Mike Haven]

this is the way i do it, in simple English instead of by mathematical formulae, which are usually gobbledegook to me, i'm ashamed to have to admit

you know 5 cards

there are 47 unknown cards of which 9 are yours, so the fraction of the deck that fills your flush on the turn is 9/47ths

to find a percentage you multiply a fraction by 100, so (9/47) x 100 gives you the 19.15% chance of filling on the turn you mention

if you miss, you know 6 cards, and there are now 46 unknown cards, of which 9 are yours

the fraction is 9/46ths, or 19.57%

i think the step you are missing is that if you fail to flush on the turn you have to say that of the (100 - 19.15)% = 80.85% of time you fail you will then succeed on the river 19.57% of this time

19.57% of 80.85% is 80.85% x (19.57/100) = 15.82%

so you hit 19.15% + 15.82% of the whole time, which is 34.97% of the whole time

[Vulpine]

The probability of making the flush by the river plus the probability of not making the flush by the river must add to 1. It’s easier to calculate the probability of not making the flush on two trials because these are independent events.

The probability of not making a flush by the river is:

Probability of not making flush on turn * probability of not making flush on river

Which is:

(47 unseen cards – 9 cards making a flush)*(46 unseen cards – 9 cards making a flush)

Probability of making the flush by the river = 1 – Probability of not making the flush by the river

Hence:

The probability of making the flush by river = 1 – (38/47)*(37/46) = 34.967


This is a classic problem that was originally solved by both Blaise Pascal and Pierre de Fermat and set forth the modern theory of probability. See page 24 of the:

"American Mensa Guide To Casino Gambling: Winning Ways"
by Andrew Brisman, Sterling Publishing; (December 31, 1999)
ISBN: 080694837X