fluctuations in expected flop distribution

the chance of flopping two cards of the same suit should be roughly 47%, if my math is correct. (flopping 3 of the same suit should occur roughly 5.3% of the time) i have been playing at a site where the flop has a flush draw of two or three cards ~60.12% over the last 2,000 games in which i have participated. at what point should i become alarmed by this departure from the expected rate?

i would remark that this anomaly has been remarkably stable, in that relatively small samples duplicate these values.

[Tom1975]

Your math's wrong. The odds of a non-rainbow flop are:

1 - (39/51)*(26/50)=60.23%

[SunOfBeach]

the problem isnt the site, its your math.

the chance of a flop with two or more suited cards is as follows: start with 1, take away the chance of a nonsuited second card (which is 12 from the remaining 51 cards = 39/51), then take away the probability of a nonsuited third card (which is 24 from the remaining 50 cards = 26/50). rounding to estimate this since i dont have a calculator handy, you get 1 - (40*25)/2500 = 1 - 0.4 = 60%, right what youre observing on your site.

that's counter-intuitive: if you have three holes, and 52 marbles of four distinct colors to fill them, you will more often have three colors than 2 or 1, reaching blindly in a bag to grab a marble, isnt that so? what am i missing?

[SunOfBeach]

[ QUOTE ]
that's counter-intuitive: if you have three holes, and 52 marbles of four distinct colors to fill them, you will more often have three colors than 2 or 1, reaching blindly in a bag to grab a marble, isnt that so? what am i missing?

[/ QUOTE ]

no. P(3) < P(2) + P(1).

do you really think you see a rainbow flop more than half of the time!!?!? thats not counter-intuitive at all, at least not counter to my intuition.

[SumZero]

[ QUOTE ]
that's counter-intuitive: if you have three holes, and 52 marbles of four distinct colors to fill them, you will more often have three colors than 2 or 1, reaching blindly in a bag to grab a marble, isnt that so? what am i missing?

[/ QUOTE ]

That isn't so. Assuming you have 13 marbles of each color (say RGBW), than consider what happens when you are picking the third marble when the first two are of different colors. Say you already have a R marble and a G marble. Now there are 26 marbles that will give you 3 different colors and 24 marbles that will give you a two colored set of three marbles. So once you have two different colored marbles you are very slightly more likely to have three different colors (26/50 or 52% of the time. So even when you start with two different color marbles you have three different colors 48% of the time which is more than the 47% you stated in your OP).

But now consider that you don't always start with two different colored marbles, some of the time the first two marbles are the same. How often does this happen? Well once you've pulled a R marble there are 39 non-R marbles and 12 R marbles so 12/51 times you will have the same color. So the correct answer for how often you have at least two of the same color is 12/51 + 39/51*24/50. That is roughly equal to 1/4 + 3/4*1/2 or 5/8 or 62%. It is actually equal to 60.235294% (to 6 decimal places accuracy).

The fact that the site in question is producing about 60% flops is good, as if it wasn't there would be something to worry about.

i see that i elided from my reasoning the immediate match of the original marble. 39/51% of the time, the second marble will not be identical to the first. the third marble has a 24/50 chance to match either the 1st or the 2nd marble. therefore, 1-39/51*24/50 of the time, there will be a match.

sorry, shoddy reasoning on my part.

what a relief, i was thinking i'd have to quit playing poker!