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tossing a coin probability calculus
A wise man once said that if you can't explain a particular thing to a seven year old child, then you haven't really understood it yourself. Now I'm not seven, but I'm hysterically afraid of maths and try to avoid anything about it. But the following contradiction (?) intrigues me so much that I just have to ask about it. Please be gentle when you explain it.
Ok, here goes. I've heard two different statements about tossing a coin, and I can understand and accept each of them separately, but it seems to me they contradict each other. Don't they? Or is it that one of them's false?
1) When you toss a coin, each time is a new time, and the chances are always 50/50 for both heads and tails, independently of all previous occassions. (Or perhaps I should say "either heads or tails", but hey you know what I mean. I'm from Sweden, goddammit.)
2) If you toss a coin a thousand times, the likely outcome is about fivehundred heads and fivehundred tails.
Now, how can the coin 'know' whether or not it's part of a thousand-toss-series...? Do you understand my logical problem here? I would really like to understand this, so please don't be afraid to sound condescending if you explain it in very simple terms! Thanks in advance, I have trouble signing in and so it may be a while before I get to thank you! (But I will certainly read your answers, hungrily.)
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The two statements are entirely consistent with each other, and both of them are true and accurate. I don't see any contradiction. You don't have anything to worry about.
The coin does not "know" anything about how many heads or tails it has already done, and that is why the total averages out in the long run at about 50% each.
The coin does not "know" anything about how many heads or tails it has already done, and that is why the total averages out in the long run at about 50% each.
both are correct.
flipping a coin is known as an "independant event" by the statisticians. what this means is that the coin does not know the outcome of the previous coin flips, or indeed the next coin flip.
so when you flip a coin (assuming no other interventions here, like a slightly heavier tails side, or something else that would/could affect the outcome), there is a 50% chance of you getting tails, and 50% chance of you getting heads. i.e. there is an equal chance of both.
so if you flip the coin 10 times, then on average you will get 5 heads and 5 tails, as they both have equal chances of happening.
if you flip it 100 times, then you will, on average, get 50 heads and 50 tails. i say on average here, because there is a chance that you may get 100 tails, or 100 heads. very unlikely, but possible.
an I explaining it well enough?
flipping a coin is known as an "independant event" by the statisticians. what this means is that the coin does not know the outcome of the previous coin flips, or indeed the next coin flip.
so when you flip a coin (assuming no other interventions here, like a slightly heavier tails side, or something else that would/could affect the outcome), there is a 50% chance of you getting tails, and 50% chance of you getting heads. i.e. there is an equal chance of both.
so if you flip the coin 10 times, then on average you will get 5 heads and 5 tails, as they both have equal chances of happening.
if you flip it 100 times, then you will, on average, get 50 heads and 50 tails. i say on average here, because there is a chance that you may get 100 tails, or 100 heads. very unlikely, but possible.
an I explaining it well enough?
Given that heads or tails on any given toss have an equally likely probability, says that for 10 tosses the likely-hood of all tosses coming up all heads (or tails) is 1 in 2^10; this is 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 (see why I used only ten tosses) = 1 in 1024 times. So it can happen, it is just more likely that heads and tails happen equally but given all the other possible combinations of heads and tails even 50-50 (half heads and half tails) is only 50 out of 1024 times with 10 tosses. Also note that for an odd (1, 3, 5, etc.) number of tosses it would be impossible to have the same number of heads and tails!
For two tosses you might get HH HT TH or TT so 2 out of 4 have a probably for an equal number of heads and tails. We will skip three tosses since we know that we cannot possibly have an equal number of heads and tails with three tosses.
For four tosses, you might get HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH or TTTT so only 4 out of 16 combinations have a probably for an equal number of heads and tails. As the number of tosses increases the odds of having an equal number of heads and tails decreases in proportion to the number of other possible combinations that become available.
For two tosses you might get HH HT TH or TT so 2 out of 4 have a probably for an equal number of heads and tails. We will skip three tosses since we know that we cannot possibly have an equal number of heads and tails with three tosses.
For four tosses, you might get HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH or TTTT so only 4 out of 16 combinations have a probably for an equal number of heads and tails. As the number of tosses increases the odds of having an equal number of heads and tails decreases in proportion to the number of other possible combinations that become available.
And if I can add something (the first two seem to be perfectly correct, by the way), the bigger the sample, the more likely it is to be closer to 50%... I think, anyway (I�m not a mathamatician!).
When there are only ten flips of the coin, if the result was 6 heads and 4 tails, the ratio would be 60/40. If by chance it was 7 heads and 3 tails, it would be 70/30. And because only a few flips were outside of the �expected� five each. However, if the coin was tossed 1,000 times and only a very small number deviated from the �expected 500 each, the ratio would deviate only slightly from 50/50.
It�s still unpredictable, of course. You could have flipped the coin 99 times and only seen it land on heads four or five times (which itself would be pretty unlikely), but it would be foolhardy to bet on the next flip landing heads (because there�d only been a few, so more were �due�) or for that matter, tails (because there�d been so many, so you expect more to follow in the same way). It�s all as unpredictable as the toss of a coin!
Confused? I am!
When there are only ten flips of the coin, if the result was 6 heads and 4 tails, the ratio would be 60/40. If by chance it was 7 heads and 3 tails, it would be 70/30. And because only a few flips were outside of the �expected� five each. However, if the coin was tossed 1,000 times and only a very small number deviated from the �expected 500 each, the ratio would deviate only slightly from 50/50.
It�s still unpredictable, of course. You could have flipped the coin 99 times and only seen it land on heads four or five times (which itself would be pretty unlikely), but it would be foolhardy to bet on the next flip landing heads (because there�d only been a few, so more were �due�) or for that matter, tails (because there�d been so many, so you expect more to follow in the same way). It�s all as unpredictable as the toss of a coin!
Confused? I am!
Hm, well, I don't know how to explain exactly what it is that I just can't get my brain around... Say that I've been tossing a coin ninethousandandninetynine times and getting tails all the time. Highly unlikely, yes, but it helps me accentuate my point (no not arguing, just trying to explain what I mean.) Since it is likely that (quoting you, bernardo) the total averages out at about 50% each in the long run, it seems to one part of my brain that the chances of getting heads should increase a little for each time I've gotten those persistent tails, or else how will the average 50% each result ever come about... And to another part of my thinking, yes I do understand that both statements must be correct, of course. Sigh. And yes fo3nix you explain it well. It's me, I'm afraid... Logging out now, will return later this week.
Correction: For four tosses, you might get HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH or TTTT so only six out of 16 combinations have a probably for an equal number of heads and tails. This is still less than half of the possible combinations.
Still checking on the ten coin toss?
Hej det ar bara jag igen... Oh I mean it's just me, again. More answers had arrived while I was responding to the first two. mibn2cweus, I'm going to have to print your answer out, but I suspect it's the kind of answer I block myself to... Not your fault, mine - some kind of maths trauma early in life, you see. snook, yes you get what I mean!
Da Swede, I think there is some confusion in the way you're understanding some words that statisticians use.
In tossing an unbiased coin, the probability of landing either a head or tail is 1/2 or 50%. Now having read the statement: "if we toss a coin 1000 times, on average, we would get 500 tails and 500 heads"; you're thinking, "what if I get 500 heads in the first 500 tosses, does that mean I must get 500 tails, because somehow the coins have memory!" Ummmmmmm, no.......that would be implying coins have memory, which of course we know they don't!
When we say; on average we will get 500 tails and 500 heads, when tossing a coin 1000 times what we mean is we can EXPECT 500 tails and 500 heads. It doesn't mean we will necessarily get them.
BUT: The larger number of tosses you have, the more likely are you to see the ratio of heads to tails tend to 1/2. This is not because heads and tails somehow know this, its because we know that each toss has a 50% chance of head or tail and even if we got 100 tails initially, the chance of the next toss being a head or tail is still 50%. Thus with lots and lots of tosses we can "expect" the ratio to come out to a half.
Summary: Because the chance of a head or tail is 50% irrespective of whats happened before, we can expect 50% of the tosses to be heads and 50% tails. The more tosses we do in our experiment, the mroe likely the ratio of heads and tails would tend to a 1/2.
I hope that makes more sense.
In tossing an unbiased coin, the probability of landing either a head or tail is 1/2 or 50%. Now having read the statement: "if we toss a coin 1000 times, on average, we would get 500 tails and 500 heads"; you're thinking, "what if I get 500 heads in the first 500 tosses, does that mean I must get 500 tails, because somehow the coins have memory!" Ummmmmmm, no.......that would be implying coins have memory, which of course we know they don't!
When we say; on average we will get 500 tails and 500 heads, when tossing a coin 1000 times what we mean is we can EXPECT 500 tails and 500 heads. It doesn't mean we will necessarily get them.
BUT: The larger number of tosses you have, the more likely are you to see the ratio of heads to tails tend to 1/2. This is not because heads and tails somehow know this, its because we know that each toss has a 50% chance of head or tail and even if we got 100 tails initially, the chance of the next toss being a head or tail is still 50%. Thus with lots and lots of tosses we can "expect" the ratio to come out to a half.
Summary: Because the chance of a head or tail is 50% irrespective of whats happened before, we can expect 50% of the tosses to be heads and 50% tails. The more tosses we do in our experiment, the mroe likely the ratio of heads and tails would tend to a 1/2.
I hope that makes more sense.
Da Swede, I think there is some confusion in the way you're understanding some words that statisticians use.
In tossing an unbiased coin, the probability of landing either a head or tail is 1/2 or 50%. Now having read the statement: "if we toss a coin 1000 times, on average, we would get 500 tails and 500 heads"; you're thinking, "what if I get 500 heads in the first 500 tosses, does that mean I must get 500 tails, because somehow the coins have memory!" Ummmmmmm, no.......that would be implying coins have memory, which of course we know they don't!
When we say; on average we will get 500 tails and 500 heads, when tossing a coin 1000 times what we mean is we can EXPECT 500 tails and 500 heads. It doesn't mean we will necessarily get them.
BUT: The larger number of tosses you have, the more likely are you to see the ratio of heads to tails tend to 1/2. This is not because heads and tails somehow know this, its because we know that each toss has a 50% chance of head or tail and even if we got 100 tails initially, the chance of the next toss being a head or tail is still 50%. Thus with lots and lots of tosses we can "expect" the ratio to come out to a half.
Summary: Because the chance of a head or tail is 50% irrespective of whats happened before, we can EXPECT 50% of the tosses to be heads and 50% tails.
I hope that makes more sense.
In tossing an unbiased coin, the probability of landing either a head or tail is 1/2 or 50%. Now having read the statement: "if we toss a coin 1000 times, on average, we would get 500 tails and 500 heads"; you're thinking, "what if I get 500 heads in the first 500 tosses, does that mean I must get 500 tails, because somehow the coins have memory!" Ummmmmmm, no.......that would be implying coins have memory, which of course we know they don't!
When we say; on average we will get 500 tails and 500 heads, when tossing a coin 1000 times what we mean is we can EXPECT 500 tails and 500 heads. It doesn't mean we will necessarily get them.
BUT: The larger number of tosses you have, the more likely are you to see the ratio of heads to tails tend to 1/2. This is not because heads and tails somehow know this, its because we know that each toss has a 50% chance of head or tail and even if we got 100 tails initially, the chance of the next toss being a head or tail is still 50%. Thus with lots and lots of tosses we can "expect" the ratio to come out to a half.
Summary: Because the chance of a head or tail is 50% irrespective of whats happened before, we can EXPECT 50% of the tosses to be heads and 50% tails.
I hope that makes more sense.
The odds of getting half heads and half tails for two tosses is 50%.
The odds of getting half heads and half tails for four tosses is 37.5%.
The odds of getting half heads and half tails for six tosses is 31.25%.
The odds of getting half heads and half tails for eight tosses is 26.5625%.
The odds of getting half heads and half tails for ten tosses is 18.75%.
The odds of getting 500 heads and 500 tails for 1000 tosses is not too good?
Ok, heres my 2 pence worth (hehe), the expected outcome would be 500/500. But that is just using the statistics to prove the most likely single outcome. If we actually tossed this coin 1000 times then chances of getting 500/500 are phenomenally small, what with all the other possibilites: 499/501, 750/250, 0/1000. But the single most likely outcome is 500/500 because the odds of either one in a single toss are 50% each.
Also you can consider it this way. If you tossed the coin 1000 times and recorded which the result was for the first, second, third toss, etc. and the answer was H,T,H,H,T,H,H(whatever)............ Then after this event the chances of repeating this sequence are *really* small. But before the event, then chances of this sequence are just as likely as all the other sequences.
I think what you are doing is looking at the sequence from the perspective of after the event and expecting to get 500/500, which is unlikely: as each toss is individual. But before the event, whilst it is no more certain, the odds predict that it will be the result because the chance of either side is 50%: any other answer would be the expression of a guess not mathematics.
Also you can consider it this way. If you tossed the coin 1000 times and recorded which the result was for the first, second, third toss, etc. and the answer was H,T,H,H,T,H,H(whatever)............ Then after this event the chances of repeating this sequence are *really* small. But before the event, then chances of this sequence are just as likely as all the other sequences.
I think what you are doing is looking at the sequence from the perspective of after the event and expecting to get 500/500, which is unlikely: as each toss is individual. But before the event, whilst it is no more certain, the odds predict that it will be the result because the chance of either side is 50%: any other answer would be the expression of a guess not mathematics.
No one seems to have mentioned the key point. Ratio, the larger the sample to closer you get to 50%, the actual amount of each is irelevant. Ignoring the edge pedants ok!? The outcome get's ever closer to 50-50 as the sample size tends to infinity. So the bigger the sample the closer to 50-50 you get. Small samples can distort, that's why you can get a run of Black or red in roulette.
Loosehead: yea, I was thinking of saying that myself, but decided not to as thought it may confuse a little.
DaSwede: to get past this infinity issue Loosehead has brought up, imagine a monkey sat at a typewriter. Is it likely to type out the complete works of Shakespere? no, not very likely. You'll most probably just get a load of rubbish, something like "dfgljfshi" right? Of course though, it IS possible for the monkey to type out all out Shakespere, just incredibly unlikely (you'll probably have to wait longer than the universe has so far existed just to see it happen).
What if you have two monkeys? More likely, but it still most probably won't happen. What about 100 monkeys? Even more likely, but still probably won't happen. What about 10,000 monkeys? Even more likely!! But still, chances are that you'll just get lots and lots of "dfkgdflkgj", and no Shakespere.
However, if you had an infinite amount of monkeys, then you would get the complete works of shakespere! You'd get an infinite number of copies of it too! And everything else ever written by man. Of course, you can never actually have an infinite amount of something, it's just a mathematical idea. So sadly, there is no guarantee that you'll get the complete works of shakespere, no matter how many monkeys you have.
DaSwede: to get past this infinity issue Loosehead has brought up, imagine a monkey sat at a typewriter. Is it likely to type out the complete works of Shakespere? no, not very likely. You'll most probably just get a load of rubbish, something like "dfgljfshi" right? Of course though, it IS possible for the monkey to type out all out Shakespere, just incredibly unlikely (you'll probably have to wait longer than the universe has so far existed just to see it happen).
What if you have two monkeys? More likely, but it still most probably won't happen. What about 100 monkeys? Even more likely, but still probably won't happen. What about 10,000 monkeys? Even more likely!! But still, chances are that you'll just get lots and lots of "dfkgdflkgj", and no Shakespere.
However, if you had an infinite amount of monkeys, then you would get the complete works of shakespere! You'd get an infinite number of copies of it too! And everything else ever written by man. Of course, you can never actually have an infinite amount of something, it's just a mathematical idea. So sadly, there is no guarantee that you'll get the complete works of shakespere, no matter how many monkeys you have.
When it comes to monkey business I may be only slightly better but I have found a new key on my calculator; it give the permutation of how many of a specific combination is possible in a given number of coin tosses. Example: For 10 tosses their are 1024 possible outcomes; of these 252 would yield 5 heads and 5 tails; (less than 25%).
For 1000 tosses the number of possible outcomes is 2x2x2...1000 times or about 10715086071900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.0, of these only about 270288240945000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.0 would yield an equal number (500 each) of heads and tails; about two and one/half percent. This is still much more likely that the possibility that all would be heads or tails. Out of all those possible combinations only 1 results in all heads (or all tails).
For 1000 tosses the number of possible outcomes is 2x2x2...1000 times or about 10715086071900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.0, of these only about 270288240945000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.0 would yield an equal number (500 each) of heads and tails; about two and one/half percent. This is still much more likely that the possibility that all would be heads or tails. Out of all those possible combinations only 1 results in all heads (or all tails).
Ok, here I am again - overwhelmed by your response. Definitely going to print all this out; thank you so much! At a quick glance through the answers, I think that you, MrPahoehoe, may have actually opened up a gate for me, by pointing out the difference between looking at a sequence in retrospect and trying to 'predict' it. As for the sample size aspect that several of you mention, yes I did understand about that, but that didn't eliminate my (perceived) logical problem. fo3nix, I find Infinite Amount of Monkeys would be a good name for a band! Well seriously folks, I would like to thank you each and every one individually here, but I won't, instead I'll print this out and study it closely. I suspect that some truly mystical aspect will continue to cling to this, for me, but I do see your points and will (most likely!) return to your 'essays' more than once. Until then, I leave you with a quote from a Swedish statis... statistician? who put it so neatly: Unlikely events are likely to occur.
(Don't remember his name, sorry.)
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