### Author Topic: A fact on probability that might blow a fuse in your brain  (Read 2075 times)

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#### jaimehlers

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##### A fact on probability that might blow a fuse in your brain
« on: January 24, 2014, 10:01:09 PM »
I think we can all agree that if you flip a coin, the chances of getting either heads or tails are 50%, no matter how many times you flip it.  But let's say you flip it, and get a tails.  If you flip it again, what are the odds of getting another tails?

If you said 50%, you would be wrong.  How is that possible?  It's because the odds of getting two tail flips in a row are only 25%, even though the odds of getting a tails result on either flip are 50%.  If you just look at a given flip in isolation, the odds of getting one specific result are going to be 50%, and never change.  But if you take a set of flips, the odds of continuing to get that specific result (or more accurately, of not getting a specific result) drop every time you flip.  It's the same reason that you only have a 1/6 chance of rolling a 6 on a six-sided dice, but you have a 11/36 chance of rolling at least one 6 if you roll two six-sided dice.

The formula for figuring it out is surprisingly simple.  It's xn/yn, where x is the number of results that aren't the one you're aiming for, y is the number of total results, and n is the number of tries you've made.  For a coin flip, it's simple; x is always going to be 1, because 1 to any power remains 1.  So it's simply dividing 1 by successive powers of 2.  For a dice roll (like that six-sided dice I just mentioned), it gets a bit trickier, because x increases along with y, but since x is always smaller than y, the ratio of xn to yn decreases as n increases.

What's even more fun is that you can use this for other things, like, say, the probability of life existing on other planets, or the probability of life occurring on its own.  Naturally, it's not realistic to figure out the exact odds, since there are several things that we can't properly quantify, but we can figure out approximate odds and then run them through the formula to get the odds of it happening over time.  I don't have access to a powerful enough calculator to actually calculate meaningfully long odds (like some of the ridiculously huge odds creationists like to give about abiogenesis happening), but I think this shows that even when the chances are initially tiny, provided you have enough opportunities for it to happen, it will eventually become very likely to happen.
Nullus In Verba, aka "Take nobody's word for it!"  If you can't show it, then you don't know it.

#### Nam

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #1 on: January 25, 2014, 12:29:24 AM »
You lost me at the math. Anyway, that's what double-headed coins are for. 100% heads all the time.

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#### Jag

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #2 on: January 26, 2014, 01:42:10 PM »
My Biology instructor walked us through this last semester, to help explain how natural selection actually works over generations. Makes perfect sense if you think it all the way through, but I wouldn't have figured it out without seeing the math and getting an explanation. Proving once again I suppose, that the "obvious" answer is not always as "obvious" as we assume at first glance.
"Tell people that there's an invisible man in the sky that created the entire universe and the majority believe you. Tell them the paint is wet, and they have to touch it to be sure." ~George Carlin

#### jaimehlers

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #3 on: January 26, 2014, 02:10:15 PM »
Well, the point was to show that probability was weird, and it is.  I mean, the odds of getting a tails on a flip is 50%, but the odds of getting two tails is only 25%, so you end up with the weird situation where the odds are two different values, both correct.  Kind of like Schrodinger's coin, in a way.
Nullus In Verba, aka "Take nobody's word for it!"  If you can't show it, then you don't know it.

#### Azdgari

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #4 on: January 26, 2014, 03:29:14 PM »
I think we can all agree that if you flip a coin, the chances of getting either heads or tails are 50%, no matter how many times you flip it.  But let's say you flip it, and get a tails.  If you flip it again, what are the odds of getting another tails?

If you said 50%, you would be wrong. ...

The rest of your post is spot on, jaime, but your way of wording this isn't quite accurate.  If you've already flipped and gotten a tails, then that probability of getting tails is now 100% - not 50%.  Because it's already happened.  Getting tails again has a probability of 50%, because it's 50% times 100% instead of 50% times 50%.
I always say what I mean. But sometimes I'm a sarcastic prick whose tone can't be properly communicated via text.

#### Jag

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #5 on: January 26, 2014, 03:30:02 PM »
Well, the point was to show that probability was weird, and it is.  I mean, the odds of getting a tails on a flip is 50%, but the odds of getting two tails is only 25%, so you end up with the weird situation where the odds are two different values, both correct.  Kind of like Schrodinger's coin, in a way.

Agreed. It bent my brain pretty hard at first, but I (metaphorically) gritted my teeth and hung with the instructor through the explanation. It was kind of a "light bulb" moment for me when it all came together. I was in the "50% odds on every flip, duh" camp until he demonstrated exactly why it was wrong.

I like it when someone "proves me wrong" and shows me a flaw in my thinking process - it shows me that I'm I'm actively learning things that matter and improving my ability to think well.

"Tell people that there's an invisible man in the sky that created the entire universe and the majority believe you. Tell them the paint is wet, and they have to touch it to be sure." ~George Carlin

#### jaimehlers

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #6 on: January 26, 2014, 04:07:27 PM »
The rest of your post is spot on, jaime, but your way of wording this isn't quite accurate.  If you've already flipped and gotten a tails, then that probability of getting tails is now 100% - not 50%.  Because it's already happened.  Getting tails again has a probability of 50%, because it's 50% times 100% instead of 50% times 50%.
That was what I like to think of as an attention-getter, actually.  But that's why I acknowledged that the odds don't actually change.  Each flip is not affected by any that came before it, or any that come after.  So what's really being calculated (to get the 25%) is the odds of having gotten a tails (which is still a probability, even though the flip has already happened) and then getting another tails after that.  That's why it's 50% and 25% at the same time.
Nullus In Verba, aka "Take nobody's word for it!"  If you can't show it, then you don't know it.

#### Azdgari

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #7 on: January 26, 2014, 05:31:48 PM »
One is predictive (50%), the other is reflective (25%).
I always say what I mean. But sometimes I'm a sarcastic prick whose tone can't be properly communicated via text.

#### magicmiles

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #8 on: January 26, 2014, 06:00:35 PM »
Reminded me of this, not sure if you saw it originally Jaime

http://whywontgodhealamputees.com/forums/index.php/topic,21218.0.html
It's good to know the door can still be open wide.

#### jaimehlers

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #9 on: January 26, 2014, 06:30:24 PM »
I've heard of the Monty Hall problem.

There is a scenario where the odds actually do change, though - it's when you do multiple flips (or rolls) at the same time.  If you flip two coins, for example, the odds of getting both heads or both tails are 25%, and the odds of getting one of each are 50%.
Nullus In Verba, aka "Take nobody's word for it!"  If you can't show it, then you don't know it.

#### xyzzy

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #10 on: January 26, 2014, 08:39:33 PM »
The thing, though, with the Monty Hall problem is that the role of the host (Monty Hall) affects the outcome in a way that is often missed or misunderstood.

In all, it's a wonderful example of an ass-backwards counter-intuitive example of probability, but it's really - really - important to understand the effect the host has on the problem.

[wiki=Monty_Hall_problem]Monty Hall Problem[/wiki]
The first principle is that you must not fool yourself - and you are the easiest person to fool -- Richard Feynman

You are in a maze of twisty little religions, all alike -- xyzzy

#### Mrjason

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##### Re: A fact on probability that might blow a fuse in your brain
« Reply #11 on: January 27, 2014, 05:35:43 AM »

There is a scenario where the odds actually do change

...like when you put an accumulator bet on