Coin Tossing are never 50/50 and ...

Googled this, interesting, cheers.

*As always a lot of the "Trick" is in making sure the wording is exactly right. (Which it is)

**In fact I'm not keen on the wording the more I look at it as you can correctly get the wrong answer or the right one depending on how you interpret it.

I would add the words "Starting from scratch" to each of the three scenarios.

A but that doesn't change the probability of a head or tail, it's just the way the problem is set out with an overlapping sequence. 

The article is an illustration that you can bias a toss

http://en.wikipedia.org/wiki/Penney%27s_game

2 different ideas really. The physics of tossing a small engraved disc and the probability of sequences of 3, that consist of the same 2 things at random

Of course 50/50 is not quite true because there is a very little chance of a coin dropping and staying on its edge :p

ok A) is the correct answer

why?

because HTH overlaps itself and what that means is that you can have this sequence appear with less coin tosses

HTHTH = you can have five tosses to achieve two sequences

now you cant ever do that with HTT

HTTHTT = you need six tosses to achieve two sequences

A. The average number of tosses until HTH is larger than the average number of tosses until HTT

because HTH overlaps itself and what that means is that you can have this (HTH) sequence appear with less coin tosses

have you got it back to front?

LOL coming to think of it, YES!!

where you read "tosses" you should read "occurrences"

thanks breandauk1

A. The average number of tosses until HTH is larger than the average number of tosses until HTT

because HTH overlaps itself and what that means is that you can have this (HTH) sequence appear with less coin tosses

have you got it back to front?

Not according to the Youtube clip. HTH on average takes 10 spins, HTT takes 8 spins

no, he himself got confused and said HTT 8 spins but in the end corrected to HTH (6:32), also in the powerpoint presentation you can see it clearly HTT 10th toss (4:17)

Tossing a coin is a complete waste of time .. Two guys  could sit down for the next 10 years  7x7 365 days .. and there would not be 40 points in it  it is evens , it  could be  evens  .. have you guys got 20 years to waste .. try the horses

you can see it clearly HTT 10th toss (4:17)

Think that is for that particular sequence HHTHHTHHTT not on average

Two guys  could sit down for the next 10 years  7x7 365 days .. and there would not be 40 points in it

Not true zipper. The more throws there are the bigger the expected gap between the two.

Feck ,who told you that ?

The theory of probability becomes of enhanced value to gamblers when it is used with the law of large numbers. The law of large numbers states that:

“If the probability of a given outcome to an event is P and the event is repeated N times, then the larger N becomes, so the likelihood increases that the closer, in proportion, will be the occurrence of the given outcome to N*P.”

For example:-

If the probability of throwing a double-6 with two dice is 1/36, then the more times we throw the dice, the closer, in proportion, will be the number of double-6s thrown to of the total number of throws. This is, of course, what in everyday language is known as the law of averages. The overlooking of the vital words 'in proportion' in the above definition leads to much misunderstanding among gamblers. The 'gambler's fallacy' lies in the idea that “In the long run” chances will even out. Thus if a coin has been spun 100 times, and has landed 60 times head uppermost and 40 times tails, many gamblers will state that tails are now due for a run to get even. There are fancy names for this belief. The theory is called the maturity of chances, and the expected run of tails is known as a 'corrective', which will bring the total of tails eventually equal to the total of heads. The belief is that the 'law' of averages really is a law which states that in the longest of long runs the totals of both heads and tails will eventually become equal.

In fact, the opposite is really the case. As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger.

Let us return to our example of 60 heads and 40 tails in 100 spins, and imagine that the next 100 spins result in 56 heads and 44 tails. The 'corrective' has set in, as the percentage of heads has now dropped from 60 per cent to 58 per cent. But there are now 32 more heads than tails, where there were only 20 before. The 'law of averages' follower who backed tails is 12 more tosses to the bad. If the third hundred tosses result in 50 heads and 50 tails, the 'corrective' is still proceeding, as there are now 166 heads in 300 tosses, down to 55-33 per cent, but the tails backer is still 32 tosses behind.

Put another way, we would not be too surprised if after 100 tosses there were 60 per cent heads. We would be astonished if after a million tosses there were still 60 per cent heads, as we would expect the deviation from 50 per cent to be much smaller. Similarly, after 100 tosses, we are not too surprised that the difference between heads and tails is 20. After a million tosses we would be very surprised to find that the difference was not very much larger than 20.

A chance event is uninfluenced by the events which have gone before. If a true die has not shown 6 for 30 throws, the probability of a 6 is still 1/6 on the 31st throw. One wonders if this simple idea offends some human instinct, because it is not difficult to find gambling experts who will agree with all the above remarks, and will express them themselves in books and articles, only to advocate elsewhere the principle of 'stepping in when a corrective is due'.

It is interesting that despite significant statistical evidence and proof of all of the above people will go to extreme lengths to fulfill there belief in the fact that a corrective is due. The number 53 in an Italian lottery had failed to appear for some time and this lead to an obsession with the public to bet ever larger amounts on the number. People staked so much on this corrective that the failure of the number 53 to occur for two years was blamed for several deaths and bankruptcies. It seems that a large number of human minds are just simply unable to cope with the often seemingly contradictory laws of probability. If only they had listened to their maths teacher. The full story is publish here.

An understanding of the law of the large numbers leads to a realisation that what appear to be fantastic improbabilities are not remarkable at all but, merely to be expected.

Lusitano ,
just read that  the number  of tosses  the more it is evens
zip told  you that .. and zip did not go to school

Lusitano  .. see my post  16.05

You obviously didn't read Lustiano's post zipper.

In fact, the opposite is really the case. As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger.

Feck, dont take this wrong  I know the odds inside out .. if  Lusitano 71 .. has a  different view .. well good luck to  her /him   its all  about opinions .. and my opinion is as good  as anyones . and i put my money were my mouth is .. unlike some  thats why iam   know   as zipper
Zip It Zipper  thats the wife . dont you just love em .. she reads this now and again  Sue if you read this  .....I Love You

zipper, you really must let me know who your scriptwriter is.

jesus wept. there is none so blind as he who will not see.

In fact, the opposite is really the case. As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger.

It seems to me there must be some point when the gap gets smaller, for instance
60/40 followed by 45/55..

Indeed Kenilworth. Who is to say that in the next 100 tosses of the coin, it does not even out to 40/60 and you are much closer to 50/50 again?

Zipper: Two guys  could sit down for the next 10 years  7x7 365 days .. and there would not be 40 points in it.

Feck N. Eejit
Not true zipper. The more throws there are the bigger the expected gap between the two

Zipper is correct, as a percentage (and that's what matters here) the gap will close with the more throws there are, but obviously the gap will usually be larger as a number simply because of the amount of throws is larger as a number.

The best of 10 throws could result in 2 heads and 8 tails, a difference of 6 or 300%, but you won't throw 1000 times and get a difference anywhere near that large even though the difference as a number may well be 60 not 6, which means the gap has closed.

I don't buy in to the HTH / HTT theory either, it's just a play, the chance of either is 7/1, there is no adavantage to calling HTH over HTT.

There's no theory to buy into, it's correct. The wording is what makes it look odd though.

If you and I start of flipping coins I'll call HTT and I'll hit my HTT first more often than you hit your HTH. If we then start again, I have an edge.

This is because there are two ways for HTx to pan out

HTH and HTT, obviously.

So, we both get to HTx after two tosses and lose

You've flipped:

HTT

and I've flipped

HTH

so we keep flipping as nobody has won

However, I'm already 1/3rd of a way to winning while you're restarting



The reason it "works" is that if we have a constant stream of flips, then They'll come up the same amount. If the string "HTHTH" comes up in a string then HTH has come up twice. By the rules of the game we're playing though, when you get HTH you win, and we start again, so the "TH" fourth and fifth flips never get you that extra victory.

the chance of either is 7/1

At the start of the game, this is correct for the first three flips.
If we both flip a heads, now we're both 3/1 to hit in the next two.
If we now both flip a tails we're both evens to hit next go.

Of those evens shots.

If we both hit = draw
If you hit and I miss = you win
If I hit and you miss = I win
If we both miss = keep flipping.

HOWEVER

We keep flipping with me starting on a H and you flipping on a T, so I'm 3/1 to hit in my next two spins, but you're back to no chance in two spins and 7/1 in the next three.

If we don't flip independently, we're 50-50 and that's my problem with the wording.

*I haven't watched the video, which may be helping

Ok i'll make it more obvious then, lets say i was betting in multiple events with two outcomes and i choose a pattern (either HTH/HTT), how many bets do i need to get a good run:

HTH = 3 tosses/ 1 pattern

HTHTH = 5 tosses/ 2 patterns

HTHYHTH = 7 tosses/ 3 patterns

HTHTHTHTH = 9 tosses/ 4 patterns

HTHTHTHTHTH = 11 tosses/ 5 patterns

HTHTHTHTHTHTH = 13 tosses/ 6 patterns

and so on....

HTT = 3 tosses/ 1 pattern

HTTHTT = 6 tosses/ 2 patterns

HTTHTTHTT = 9 tosses/ 3 patterns

HTTHTTHTTHTT = 12 tosses/ 4 patterns

HTTHTTHTTHTTHTT = 15 tosses/ 5 patterns

HTTHTTHTTHTTHTTHTT = 18 tosses/ 6 patterns

and so on...

Conclusion:
for me to achieve the same good run of 6 consecutive HTH with HTT i would need to make 5 more bets

The likelihood of HTH appearing in a random string is more often than HTT, which has statistically significance.

My tuppence flip: Assuming the coin is balanced, and a fair test, surely B, equal?

I still don't buy in to it, because you're using the previous now dead spin to claim a win.

In your example above of HTH = 13 tosses/6 patterns, I see 13 tosses = 2 patterns of HTH and 2 of THT, THT always following HTH.

How can you use a dead spin HTH in the real world to your advantage?

I don't buy into it for the simple reason no coin toss has any previous knowledge of any previous coins toss, the absolute fundamental thing about probability that some don't understand, and from that viewpoint, each sequence begins anew with each toss, and therefore each sequence has a likely probability, AS LONG AS every event in the sequence has a likely probability.

Lori : If you and I start of flipping coins I'll call HTT and I'll hit my HTT first more often than you hit your HTH. If we then start again, I have an edge.
Lori, I wouldn't normally nit pick, but you did say in another thread when I cussed off a nit picker for picking you up that it was the right thing to do :)

Lusitano71
LOL coming to think of it, YES!!

where you read "tosses" you should read "occurrences"

thanks breandauk1

Whilst in nit picking mode, what's the difference between tosses and occurences? I would have thought answer C best fits your theory, as Brendan suggested?

you guys are thinking in terms of the base pattern (3 letters) and im thinking of a pattern that keeps appearing on a never ending string of random H's and T's, simples

why should the last H be the ending and not the start of the next pattern???

Because the last H is a done deal, it's expired, you can't use it again in the real gambling world. Or if you can, how?

Can people who don't like this show me which point they disagree on?

1) There are 64 possible starts to the game.
2) One of those is the instance where both players win (Let's call that a draw?)
3) In the 16 instances where both players start with a "T" the game is even, so we can discard those
4) In the 32 instances where one player starts with a head and the other starts with a tail, the player who starts with the head has a clear advantage. As it's 50-50 who gets that advantage we can ignore those
5) In the 8 instances where one player starts HT and the other starts HH, the HT has a clear advantage. so we can ignore those
6) In the 4 instances where both players start HH, we start the game again.
7) In the 1 instance where both players start HTH , the HTH player wins
8) In the 1 instance where both players start HTT, the HTT player wins.

This leaves the instance where the player needing HTH starts HTT and the player needing HTT starts HTH.

Which gives the situation where player a) Needs HTH to win and player b) Needs TT to win.


Lustiano's posts explain where the "missing" HTH's go (They never get played out to make up their average)

Let's play an easier game with fewer combinations:

Player a) has to throw TT to win, Player b) has to throw TH to win.

If they both throw a T on their first flip, there are four outcomes 

1: a) flips T  b) flips T - player a wins
2: a) flips H  b) flips H - player b wins
3: a) flips T  b) flips H - both players win
4: a) flips H  b) flips T - see below

The state of play after two flips in scenario 4: is

a) has flipped TH
b) has flipped TT

with a third flip to come. No matter what a) flips on his next flip, his two possible chains are

THT or THH (Needing TT to win remember) so he cannot win

however:

b's possible chains are

TTH or TTT (Needing TH to win) The first of which is a winner.

Working with this case should make it easier to see.

People are confusing two different problems in this thread, overlapping sequences don't come into the original question, it's simply the person who hits his sequence 1st which will be only average HTT, hence A is the answer

Zipper: Two guys  could sit down for the next 10 years  7x7 365 days .. and there would not be 40 points in it.

Feck N. Eejit: Not true zipper. The more throws there are the bigger the expected gap between the two

Trevh: Zipper is correct, as a percentage (and that's what matters here) the gap will close with the more throws there are, but obviously the gap will usually be larger as a number simply because of the amount of throws is larger as a number.

The best of 10 throws could result in 2 heads and 8 tails, a difference of 6 or 300%, but you won't throw 1000 times and get a difference anywhere near that large even though the difference as a number may well be 60 not 6, which means the gap has closed.

In what way is zipper correct? He's implying that if they toss coins for £1 continually for 10 years one of them being £40 up is a maximum?

it's simply the person who hits his sequence 1st which will be only average HTT, hence A is the answer

Agree, there are no "dead" spins as such HTT has simply better chance of appearing first as a "failure" in the last T result is a "success" on the way to completing the sequence.

Look up pennys game which takes it a lttle further. Any sequence of 3 coins can be beaten by chosing another sequence. It was on Derren Brown to, one of the few times he did essentially a science experiment

Slightly off the point of this thread, but of course, really there are three possible outcomes: head, tails and edge (though the last is almost zero for ordinary coins).

You might wonder how thick a coin has to be for the probability of it landing on its edge to be 1/3. The mathematician John von Neumann famously calculated this in his head in about 20 seconds (obviously has to do with the position of the centre of gravity of the coin).

There are more than three possible outcomes.

-During the flip the coin could be lost, in which case the outcome is unknown.
-The coin could shatter or break on impact, or be otherwise damaged during the toss.
-The coin could land on a surface that is bumpy, making it impossible to determine whether it has landed on edge or heads/tails.
-The coin may not come to rest during the timeframe required.

- A leaping Jack Russell might catch the coin in its mouth
- An unlikely quantum effect might make the coin disappear
- You might be in space and it'll keep on going until dragged into and vapourised by a star

We could go on :)

lusitano 71   the old man zip  hear , you have  been on my thread .
now listern up son  this tossing a coin  is a waste of time .. its evens heads or tails  for  the next 100 years try the horses  zip goes all in  Advertisement  4.15.   its up there in my post  Its only a bit of Fun   it won at 11/4   lay me  11/4 heads or tails ....game over

the answer to the question as originally stated seems to me to be clearly the obvious one, ie b.

only when and as soon as the first two conditions are met, the game must end on the next flip, which is fifty fifty.

when they fail the first time HTT has head start on next sequence and head start everytime after. which translates to appearing first more often

only when and as soon as the first two conditions are met, the game must end on the next flip, which is fifty fifty.

They are not betting on one series of flips, (in which case clearly you're right) they are flipping independently.

that isn't specified in the original post, though.

The average number of tosses

you would need more than one flip to get an average number

It's horribly worded, as these things often are however it doesn't technically need to be as it asks for the average.

The problem comes from the brain then stopping at the point you reached (I had the same issue for a while) and concluding it's therefore the same.

What you have to do though is continue until the other one (The loser if you like) comes in, so you can answer the question about the average.
I found it easier to treat it as two people competing, but you can do it as worded. (Note that the second occurence of the winning combination is irrelevant - again, badly worded)

Looking at one chain:
On the occasions you get to HTx, then it's 50-50 who wins. However the loser takes longer to show if it's HTH than if it's HTT,so the AVERAGE time for HTT to show is less)

as you say, it's horribly worded. as originally stated, either of two answers can be right.

Tossing a coin   ok its heads or tails
this 4.05   its a toss of a coin   Master  Fiddle .. and Poungach   zips on Master  Fiddle  at 7/4  if you smart  gamblers , lay me 7/4  on the toss of a coin   zip will take you for The Bank Of England
Master Fiddle  ....get on

Tails won Poungach, and  it was never in doubt ..zip pays with a smile  good night  bring on the nurse  zzzzzzzz

That's the conclusion I reached too pres.

reading some of the above posts.. surely if i toss a coin once ther is no way of acheiving a pattern because the next toss result has absolutely no relevance to the previous toss.. so how is it safe 2 acheive a pattern??