Odds on a penny flip - 1 in 3?

[DrFaroohk]

A great many years ago, a math teacher taught me through some fucked up equations that if you flip a coin, the odds of it being heads are actually 1 in 3. He showed me how it worked out, and I never got it then and I don't get it now, but I don't even remember how he proved it.

Any math wizards here that know of something like this?

[thansal]

My knowledge on the subject (take from the Internets) is that your teacher was wrong (or you are possibly miss-remembering).


A flipped coin (traditional flip, like you see in any movie/tv show) has a higher probability of landing with the initial face up side still face up.

The web page:
http://www.codingthewheel.com/archiv...ir-proposition

The rational behind this is that when you flip the coin it has 2 likely states, Heads up or Tails up. Let us assume that you start with Heads up, this means that for every state heads has been up either more often or equally often as tails, thus heads has a higher probability.

A good way of thinking about this is by looking at the ratio of odd numbers to even numbers when you start counting from 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

No matter how long you count, you'll find that at any given point, one of two things will be true:

You've touched more odd numbers than even numbers
You've touched an equal amount of odd numbers and even numbers

What will never happen, is this:

You've touched more even numbers than odd numbers.

Similarly, consider a coin, launched in the "heads" position, flipping heads over tails through the ether:

H T H T H T H T H T H T H T H T H T H T H T H T H

At any given point in time, either the coin will have spent equal time in the Heads and Tails states, or it will have spent more time in the Heads state. In the aggregate, it's slightly more likely that the coin shows Heads at a given point in time—including whatever time the coin is caught. And vice-versa if you start the coin-flip from the Tails position.

Now, they also point out some other important bits:

1) Most people slap the coin on the back of their hand, thus inverting the probability.
2) It is VERY possible to control a flip by controlling the starting position and the height of the flip. I used to be able to do this (I haven't practiced in a while).
3) A SPUN coin has a HUGE weighting to it (pun intended), and the heavier side will end up down most often. This might have been where heads is 1 in 3 instead of 50/50.

HTH

[Greenday]

Well, if you count landing on it's side as an option, the probability is 1/3.

If you don't, statistically it can either come heads or tails thus it's 1/2.

[AdminAssistant]

I remember reading something about this in a philosophy book, actually, using the opening scene from Rosencrantz and Guildenstern are Dead as an example. I believe it was a debate...does the previous outcome of a flip influence the outcome of a new flip? Or do the odds reset at 50/50 for each new flip?

One of those questions that makes me a bit dizzy.

[Greenday]

I remember reading something about this in a philosophy book, actually, using the opening scene from Rosencrantz and Guildenstern are Dead as an example. I believe it was a debate...does the previous outcome of a flip influence the outcome of a new flip? Or do the odds reset at 50/50 for each new flip?

One of those questions that makes me a bit dizzy.

Just depends on how you word a question. What are the odds of flipping a quarter heads twice? 1/4. What are the odds of flipping a quarter heads after you just flipped it heads the last time? 1/2

[DrFaroohk]

Well from what I remember, it's something like a bit of wordplay. It was like "Heads OR tails", and since he didn't specify, it somehow went to 1/3. I dunno. I never learned statistics very well.

[Kogarashi]

The only weird coin probability thing I remember hearing is that most coins are slightly weight-biased toward one of their two faces, with the exception of dimes (US coinage here). Don't remember where I heard that one either.

[thansal]

The only weird coin probability thing I remember hearing is that most coins are slightly weight-biased toward one of their two faces, with the exception of dimes (US coinage here). Don't remember where I heard that one either.

That's covered in the link I posted. It doesn't effect a flip (think about it, it's going end over end until you catch it). However it makes a spun coin (spin it on a table like a top) tend to land with tails face up (the heavier side is where it's going to end up leaning down on).

1 in 3 including edge would only be if we are ignoring physics, which we shouldn't, as it's the law. The actual probability of an edge landing is around 1 in 6000 (according to above website).

The idea that a streak increases the probability of the reverse is called the Gambler's Fallacy. The world does not pay attention to previous flips of a coin, it doesn't care that you had a statistical anomaly of flipping a coin 5,000 times and having it land on it's head each time. The next flip is still a 50/50 chance (really more of a 49/51, but whatever). Admittedly, if you do have 5000 flips land as heads, you should get your self a new flipper/new coin as the guy is probably trying to cheat you (as that probability is much higher).

People that don't understand math will argue this of course, however, they are wrong. Randomness means that you get streaks, probability means that as instances of an event approach infinity the probability evens out.

[Seshat]

I believe it was a debate...does the previous outcome of a flip influence the outcome of a new flip? Or do the odds reset at 50/50 for each new flip?

Assuming true randomness (ie, discounting the possibility that the coin is slightly heavier on one side or the other, and assuming the coin flipper generates a perfect randomness): the odds reset for each flip.

Well from what I remember, it's something like a bit of wordplay. It was like "Heads OR tails", and since he didn't specify, it somehow went to 1/3. I dunno. I never learned statistics very well.

Ah. Then it's not a statistics thing, it's a wordplay thing.

Possibility of heads: 1/2.
Possibility of tails: 1/2.

Possibility that the coin will land on either heads or tails: very very close to 1. (it IS theoretically possible that it will land on its side.)

[Chromatix]

"Heads or... some sort of fish with legs."

"When it's in the air... Edge."

"Well, whaddya know."

- Terry Pratchett, The Colour of Magic

[Imogene]

"When it's in the air... Edge."

Hurray for Rincewind's luck