Common Fallacy in Odds Calculation

I FLOPPED back to back quads starting with a pocket pair eain a money game on Planet Poker a few weeks ago.


I calculated the odds of this happening as follows:


Being dealt 1 pocket pair: 16:1

Flopping quads with that pocket pair: 407:1

Total odds of 1st hand being dealt pocket pair AND flopping quads: 6512:1 (YES, that high!)

2 hands in a row like this: 42,406,144:1


WOW! The event of a LIFETIME!


BUT WAIT! THERE IS A FLAW IN MY ODDS CALCULATIONS! I actually knew what it was before I figured this out, but can you find it? It is a similar flaw used in many calculations and people are always falling victim to it. In fact, I have never heard this flaw mentioned (that I can remember).


Here it is...42,406,144:1 IS NOT the odds for any 2 hands in a row flopping quads in the manner described. It is the odds of the NEXT TWO HANDS flopping quads in this manner. If you have already flopped quads once, the odds of the next hand doing the same in the manner indicated are but 6512:1, since the first event is already in the past.


I am correct in my thinking, but I have never tried expressing my thoughts on this. I know some people will disagree with me, BUT I AM CORRECT. If someone who understands what I am saying would please clarify my explanation, I would appreciate it greatly!

Hmmmmmmmmmmmm, this seems painfully obvious. Yuo can look at the odds of certain things in any contrived way you want- hell tomorrow is Nov 11, now lets see we've had our calendar for about 2001 years, multiply that by 365 and OH MY GOD!!!!!!!!!!! November 11 2001 is about a 73,000 to 1 shot!!!! Hey what do you know this must be a pretty rare day.


Well that made a lot of sense :-)

No great falacy here. What you have calculated is the odds of ANY two hands flopping quads before they are dealt (nothing special about the next two hands). Of course once you flop quads on a hand then the probability decreases.


It is truly a rare event for an idividual to experience this (less than once in a lifetime). It is not amazing that someone here has experienced it, and if it hadn't been flopping quads twice in a row it would have been some other rare event. Rare events happen all the time /images/smile.gif

What do you mean by "Of Course once you flop quads on a hand then the probability decreases" ?

Just that. Once you have flopped quads, the odds of making it two in a row are just the odds of flopping quads on the very next hand which is 1 in 6512.


Throughout your lifetime you will flop quads many times, and each time there will be a 1 in 6512 chance of making it two in a row. That doesn't change the fact that it only happens 1 time in 42 million hands, and you will probably never do it.

Another way to look at this is that flipping a coin heads 100 times in a row after 99 heads have been flipped is 50/50.

What I would like to know is what are the odds for the coin to come up heads twice in a row? That would help illustrate the concept.


thanks, from an inept math person.

The odds of one head comming up is 1/2. Thus the odds for two heads comming up on the NEXT TWO tosses is 1/2 * 1/2, or 1/4. Three heads on your nest 3 tosses would be 1/2 * 1/2 * 1/2, or 1/8. In fact, the odds of N heads comming up on your next N tosses are 1/(2^N), or one over two to the N power. The odds af flopping quads twice in a row (on any two given hands) is the odds of flopping it once times the odds of flopping it once. i.e.

1/6512 * 1/6512, or 1/42,406,144. Note that this is NOT the same as the odds of flopping quads twice on ANY two hands in the future. i.e. flopping quads twice in a row at some point in your life is significantly less than 1/42,406,144.


-MD

Being dealt 1 pocket pair: 16:1

Flopping quads with that pocket pair: 407:1


BUT WAIT! THERE IS A FLAW IN MY ODDS CALCULATIONS

Yes there is...

Then Probability of both is 1/17 * 1/408 = 1/6936= 6935:1

I suspect that you got your 6512:1 by multiplying 16 and 407, which is not correct.

This is correct but for non math people it might help to see all the possible outcomes of two coin flips like this


First coin 2nd coin

heads heads

heads tails

tails tails

tails heads


So there are 4 possible combinations and only 1 is heads/heads and so the odds are 1 in 4