Why do coin toss results seem contradictory?

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ForgedinHell
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Why do coin toss results seem contradictory?

Post by ForgedinHell »

Take a coin, evenly weighted. It will have a 50% chance landing heads and a 50% chance landing tails. This means if we toss it a million times, we should expect about have the results to be heads, and the other half to be tails. Otherwise, we wouldn't have 50/50 probability. That being said, what is the probability that a coin toss yields any of the following results?

(a) H, H, T, T, H, T, H, T, H, H

(b) T,T, T, T, T, T, T, T, T, T

(c) H, H, H, H, H, H, H, H, H, H

Which result is more likely to occur and how does that answer confirm that we should expect 500,000 heads to come up in a million tosses? Is there a paradox here?
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The Voice of Time
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Re: Why do coin toss results seem contradictory?

Post by The Voice of Time »

I've been questioning myself similar things. A question which often pops to my mind is "if it didn't happen then what?", and I usually end up with an answer like: "The chance that it will happen is 50%", which is like an infinitely accumulating argument as then there has to be an additional "50% chance" that there is a 50% that there is a 50% that it will happen. And so on ad infinitum. Which arithmetically really isn't a 50/50 split, and I have no idea why what's wrong with my reasoning, so I just gives up and leave the answer to the mathematicians ->

Mathematicians are clever, the speak of such things as "convergence" and so forth, and would for instance point out that the two sums, as you continue to toss the coin, converges to a 50/50 split. That's my simple answer to you, analytic number theory.
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ForgedinHell
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

The Voice of Time wrote:I've been questioning myself similar things. A question which often pops to my mind is "if it didn't happen then what?", and I usually end up with an answer like: "The chance that it will happen is 50%", which is like an infinitely accumulating argument as then there has to be an additional "50% chance" that there is a 50% that there is a 50% that it will happen. And so on ad infinitum.

But mathematicians are clever, the speak of such things as "convergence" and so forth, and would for instance point out that the two sums, as you continue to toss the coin, converges to a 50/50 split. That's my petty answer to you.
Well, you are wrong. This is not a question about convergence. Which result is more likely to occur among a, b and c with an evenly balanced coin? And how does that answer tell us that we would expect 500,000 heads to come up after a million tosses. That's the question. Convergence would deal with something like the zeno paradoxes, which, by the way, that tally guy in the Philosophy Now Forum completely screwed the pooch on in describing the math involved in the paradox. There is no limit that we are dealing with here, so it is not a convergence issue. It's straight probability theory, but it sets up things for a more interesting question involving the mathematical nature of reality.
mickthinks
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Re: Why do coin toss results seem contradictory?

Post by mickthinks »

ForgedinHell wrote:
The Voice of Time wrote:But mathematicians are clever, the speak of such things as "convergence" and so forth, and would for instance point out that the two sums, as you continue to toss the coin, converges to a 50/50 split. That's my petty answer to you.
Well, you are wrong. This is not a question about convergence.
And Voice didn't say it was about convergence, he just mentioned 'convergence' in his reply! What is wrong with you, Forgy? It's like you can't have a calm reasonable discussion with anyone - it always has to be a battle, in which you always declare yourself the winner.
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Grendel
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Re: Why do coin toss results seem contradictory?

Post by Grendel »

The probabilty for any of these happening at the outset is 1/1024 this is because there are 1024 possible outcomes, after the process begins the number of possible outcomes reduce and the probability changes. Someone didn't listen at Y7 maths class, probability is calculated by number of possible outcomes, with a coin it's always a multiple of 2.
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ForgedinHell
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

mickthinks wrote:
ForgedinHell wrote:
The Voice of Time wrote:But mathematicians are clever, the speak of such things as "convergence" and so forth, and would for instance point out that the two sums, as you continue to toss the coin, converges to a 50/50 split. That's my petty answer to you.
Well, you are wrong. This is not a question about convergence.
And Voice didn't say it was about convergence, he just mentioned 'convergence' in his reply! What is wrong with you, Forgy? It's like you can't have a calm reasonable discussion with anyone - it always has to be a battle, in which you always declare yourself the winner.
Reality declares me the winner.
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ForgedinHell
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

Grendel wrote:The probabilty for any of these happening at the outset is 1/1024 this is because there are 1024 possible outcomes, after the process begins the number of possible outcomes reduce and the probability changes. Someone didn't listen at Y7 maths class, probability is calculated by number of possible outcomes, with a coin it's always a multiple of 2.
Right, they are all equally likely outcomes. But, if it is true that ten heads is as equally likely to occur as the pattern that has both heads and tails, then why should one expect a million coin tosses to produce a result of 500,000 heads and 500,000 tails. In a milion coin tosses, the chance that a series will come up half heads and half tails is no more likely than the probability of throwing 1 million heads straight. So, how do you reconcile the expectation of 500,000 heads with the fact that such a toss is no more likely than a million heads in a row?
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Grendel
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Re: Why do coin toss results seem contradictory?

Post by Grendel »

It's very simply, it's to do with distribution, 499,999 tails to 500,001 heads is nearly 50/50 and so on. More of these nearly 50/50 probabilities are distrubuted closer to the mean than the higher and lower ends of the range, so the result will always be closer to 50/50.

The probability of throwing tails with four coins. Each outcome has an even chance of happening, but there are is only 1 TTTT outcome and 6 50/50 outcomes, so more likely.


TTTT 100/0
TTTH 75/25
TTHT 75/25
THTT 75/25
HTTT 75/25
THTH 50/50
THHT 50/50
HTTH 50/50
HTHT 50/50
TTHH 50/50
HHTT 50/50
THHH 25/75
HTHH 25/75
HHTH 25/75
HHHT 25/75
HHHH 0/100
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ForgedinHell
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

Grendel wrote:It's very simply, it's to do with distribution, 499,999 tails to 500,001 heads is nearly 50/50 and so on. More of these nearly 50/50 probabilities are distrubuted closer to the mean than the higher and lower ends of the range, so the result will always be closer to 50/50.

The probability of throwing tails with four coins. Each outcome has an even chance of happening, but there are is only 1 TTTT outcome and 6 50/50 outcomes, so more likely.


TTTT 100/0
TTTH 75/25
TTHT 75/25
THTT 75/25
HTTT 75/25
THTH 50/50
THHT 50/50
HTTH 50/50
HTHT 50/50
TTHH 50/50
HHTT 50/50
THHH 25/75
HTHH 25/75
HHTH 25/75
HHHT 25/75
HHHH 0/100
Right. Although one million coin tosses landing heads up every time is equally likely as a million tosses where half the results come up heads and the other half tails, there are far more combinations involving the possibility of 1/2 heads coming up than there are combinations of all heads coming up, which can only be done one way. So, although all heads is as likely to be the result of a combination that is 50% heads, since there are more possible strings with 50% heads, we still expect a result to come up 50% heads.

But, in the example you used, involving four tosses total, there are a possible 16 combinations. Out of those 16 combinations, there are 6 combinations that contain 2 heads and 2 tails. So, the odds of getting half the tosses heads and half tails is 6/16, which is less than 50%. So, where is the expectation of 50% heads and 50% tails coming from?
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ForgedinHell
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

Grendel wrote:It's very simply, it's to do with distribution, 499,999 tails to 500,001 heads is nearly 50/50 and so on. More of these nearly 50/50 probabilities are distrubuted closer to the mean than the higher and lower ends of the range, so the result will always be closer to 50/50.

The probability of throwing tails with four coins. Each outcome has an even chance of happening, but there are is only 1 TTTT outcome and 6 50/50 outcomes, so more likely.


TTTT 100/0
TTTH 75/25
TTHT 75/25
THTT 75/25
HTTT 75/25
THTH 50/50
THHT 50/50
HTTH 50/50
HTHT 50/50
TTHH 50/50
HHTT 50/50
THHH 25/75
HTHH 25/75
HHTH 25/75
HHHT 25/75
HHHH 0/100
I forgot to add something. If we just tossed the coin twice, then we have the following possibilities: H H; H T; T H; T T. Each possibility is equally likely, 1/4 chance of any of the four coming up. And 2 of the four possibilities have one head and one tail, so when we toss the coin twice, we do have 50% odds of heads coming up. So, why is it that the odds shifted down to 6/16 or 3/8 when we toss the coin two additional times? Shouldn't the higher number of tosses make it more likely that we end up with 50% heads, since we are eliminating the randomness associated with small sample sizes? Yet, just the opposite happened when we went from 2 to 4 tosses.
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Re: Why do coin toss results seem contradictory?

Post by mickthinks »

ForgedinHell wrote:This is not a question about convergence.
ForgedinHell wrote:So, where is the expectation of 50% heads and 50% tails coming from?
LOL
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ForgedinHell
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

mickthinks wrote:
ForgedinHell wrote:This is not a question about convergence.
ForgedinHell wrote:So, where is the expectation of 50% heads and 50% tails coming from?
LOL
What is so funny? If I hold in my hand an evenly balanced coin, and I toss it one time, the odds of heads coming up is 50%. On additional tosses, since the coin does not "remember" the prior toss, the odds should always remain 50% that heads will come up, for every toss. Yet, when we crunch the actual numbers, this is not the case. Even in the case of two tosses, there is a 50% chance that we will not have one heads and one tails come up. With four tosses, the odds are greater than 50% that we won't have two heads and two tails come up. So, how is it that the coin toss is independent of the prior tosses, when the odds are in fact affected by the prior coin tosses? And, if the prior coin tosses do affect the probability of the current coin toss, then how does that happen? How does tossing a coin that has a 50% chance on each toss of coming up with heads, give us something other than a 50% chance of coming up with heads based on repeated tosses?
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Re: Why do coin toss results seem contradictory?

Post by Grendel »

ForgedinHell wrote:
Grendel wrote:It's very simply, it's to do with distribution, 499,999 tails to 500,001 heads is nearly 50/50 and so on. More of these nearly 50/50 probabilities are distrubuted closer to the mean than the higher and lower ends of the range, so the result will always be closer to 50/50.

The probability of throwing tails with four coins. Each outcome has an even chance of happening, but there are is only 1 TTTT outcome and 6 50/50 outcomes, so more likely.


TTTT 100/0
TTTH 75/25
TTHT 75/25
THTT 75/25
HTTT 75/25
THTH 50/50
THHT 50/50
HTTH 50/50
HTHT 50/50
TTHH 50/50
HHTT 50/50
THHH 25/75
HTHH 25/75
HHTH 25/75
HHHT 25/75
HHHH 0/100
Right. Although one million coin tosses landing heads up every time is equally likely as a million tosses where half the results come up heads and the other half tails, there are far more combinations involving the possibility of 1/2 heads coming up than there are combinations of all heads coming up, which can only be done one way. So, although all heads is as likely to be the result of a combination that is 50% heads, since there are more possible strings with 50% heads, we still expect a result to come up 50% heads.

But, in the example you used, involving four tosses total, there are a possible 16 combinations. Out of those 16 combinations, there are 6 combinations that contain 2 heads and 2 tails. So, the odds of getting half the tosses heads and half tails is 6/16, which is less than 50%. So, where is the expectation of 50% heads and 50% tails coming from?
a calculation of the mean.
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Re: Why do coin toss results seem contradictory?

Post by Grendel »

ForgedinHell wrote:
Grendel wrote:It's very simply, it's to do with distribution, 499,999 tails to 500,001 heads is nearly 50/50 and so on. More of these nearly 50/50 probabilities are distrubuted closer to the mean than the higher and lower ends of the range, so the result will always be closer to 50/50.

The probability of throwing tails with four coins. Each outcome has an even chance of happening, but there are is only 1 TTTT outcome and 6 50/50 outcomes, so more likely.


TTTT 100/0
TTTH 75/25
TTHT 75/25
THTT 75/25
HTTT 75/25
THTH 50/50
THHT 50/50
HTTH 50/50
HTHT 50/50
TTHH 50/50
HHTT 50/50
THHH 25/75
HTHH 25/75
HHTH 25/75
HHHT 25/75
HHHH 0/100
I forgot to add something. If we just tossed the coin twice, then we have the following possibilities: H H; H T; T H; T T. Each possibility is equally likely, 1/4 chance of any of the four coming up. And 2 of the four possibilities have one head and one tail, so when we toss the coin twice, we do have 50% odds of heads coming up. So, why is it that the odds shifted down to 6/16 or 3/8 when we toss the coin two additional times? Shouldn't the higher number of tosses make it more likely that we end up with 50% heads, since we are eliminating the randomness associated with small sample sizes? Yet, just the opposite happened when we went from 2 to 4 tosses.
3/8 doesn't exist, throwing 50/50 with an odd number of coins is impossible. So the sequence is 2/4, 0/8, 6/16, 0/32, ?/64. 2 outcomes is not enough to calculate the term to term rule, you would need to calculate the number of 50/50 for six coins, then any number of dice may be calculable, if not 8 coins should make it certain.
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Re: Why do coin toss results seem contradictory?

Post by ForgedinHell »

Grendel wrote:
ForgedinHell wrote:
Grendel wrote:It's very simply, it's to do with distribution, 499,999 tails to 500,001 heads is nearly 50/50 and so on. More of these nearly 50/50 probabilities are distrubuted closer to the mean than the higher and lower ends of the range, so the result will always be closer to 50/50.

The probability of throwing tails with four coins. Each outcome has an even chance of happening, but there are is only 1 TTTT outcome and 6 50/50 outcomes, so more likely.


TTTT 100/0
TTTH 75/25
TTHT 75/25
THTT 75/25
HTTT 75/25
THTH 50/50
THHT 50/50
HTTH 50/50
HTHT 50/50
TTHH 50/50
HHTT 50/50
THHH 25/75
HTHH 25/75
HHTH 25/75
HHHT 25/75
HHHH 0/100
I forgot to add something. If we just tossed the coin twice, then we have the following possibilities: H H; H T; T H; T T. Each possibility is equally likely, 1/4 chance of any of the four coming up. And 2 of the four possibilities have one head and one tail, so when we toss the coin twice, we do have 50% odds of heads coming up. So, why is it that the odds shifted down to 6/16 or 3/8 when we toss the coin two additional times? Shouldn't the higher number of tosses make it more likely that we end up with 50% heads, since we are eliminating the randomness associated with small sample sizes? Yet, just the opposite happened when we went from 2 to 4 tosses.
3/8 doesn't exist, throwing 50/50 with an odd number of coins is impossible. So the sequence is 2/4, 0/8, 6/16, 0/32, ?/64. 2 outcomes is not enough to calculate the term to term rule, you would need to calculate the number of 50/50 for six coins, then any number of dice may be calculable, if not 8 coins should make it certain.
But the 3/8 odds does exist. It's no different from stating a 1/16 percent chance exists for any of the combinations of four coin tosses to come up. The odds that four coin tosses will be half heads and half tails is 3/8, not 4/8, so less then 50%. The numbers don't lie.
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