Probability - How does it work?!

Yes but if heads came up 100 times in a row the chance of me noticing and shouting this from the rooftops 100% . .

: )

Yep, that's basically the problem with people misunderstanding lottery probabilities.

47, 17, 8, 12, 2, 36 looks insignificant.

Whereas . . .

1, 2, 3, 4, 5, 6 looks significant.

And this significance is ascribed a value it doesn't have (it being more or less likely that other sequences we ascribe no significance to).

The fact that they know each other is as good as irrelevant. While I guess there could be some very slight biases (people might have some connection to a shared event). and also there may be certain dates which someone is more likely to be born on (for example if people have more sex in winter), but for this question lets assume neither of these has an effect. Let's also assume there are no leap years & leap seconds, Again, this is to simplify the mathematics.

That means that for two (or more) to share a birthday it's

1/365 + 2/365 + 3/365 + 4/365 = 10/365 = 2.7% chance.

Not quite.

The way to this out is to subtract the probability that they do not share a birthday from one. For two people, there are 364 days out of 365 in the year where they don't share a birthday, a very high probability (364/365). For three people, we multiply the probability that the first two don't share a birthday (364/365) by the number of ways out of 365 that the third person can not share a birthday with the first two (363/365). And so on, so the probability that five people do not share a birthday is (364/365)(363/365)(362/365)*(361/665) or about .973 if my calculator is working. This means that the probability that they do share a birthday is .027 or 2.7%.

Same answer but your method is flawed. It's easy to see this anyway since with larger numbers you would quickly end up with probabilities greater than one.

^yes, each specific sequence is equally likely but it depends on whether the order has to be considered.

The problem (as usually stated) is what are the chances of flipping a head 100 times in a row. Rather than in an arrangement of 100 coins what percentage of possible arrangements feature XXXXX (XXXX being whatever you are measuring).

As useless was pointing out, many individual sequences give the same number of heads and tails if that is all you are interested in.

So with 4 coin tosses the possible outcomes are:

HH --- 1/4
HT --- 1/4
TH --- 1/4
TT --- 1/4
Therefore a head and a tail (order not important) = 1/4 + 1/4 = 1/2

They are all equally likely but if the order is not important it is 2 times as likely to get a head and a tail.

Well, strictly speaking you are not twice as likely to get a head + tail (on a two coin toss) when accepting combinations in place of sequences, the results aren't influenced by the rules you are using, only your interpretation of the results, but I understand the point you are making.

If a run of 'x' coin tosses was repeated infinitely many times each possible sequence would occur

Probably, but not necessarily.

I once had a long ongoing argument with a whole studio (around 20 people) where I worked (this was when the lottery first started so probably around 1995) - based loosely around this problem, basically a guy on the radio had said, during a piece on the lottery, that it was a mistake to choose all even (or all odd numbers) as those combinations are much rarer than mixed even and odd sequences.

It literally took me a month to convince people that the idea was nonsense, so strong was people's instinct that certain numeric sequences are significant in a system like this, it actually got quite funny at times, hilariously so, the whole thing ended in me actually making 49 little balls with 'Carry On' stars replacing the numbers to try and break the connection (and significance on sequence) people can erroneously place on numbers.

I got them to argue the case that:

Hattie Jacques, Sid James, Jim Dale, Terry Scott, Windsor Davies . . . (etc etc)

. . . was more (or less) likely to come up than . . .

Kenneth Williams, Charles Hawtrey, Joan Sims, Barbara Windsor, Leslie Phillips . . . (etc etc)

. . . and to explain the system by where one Carry On film star was more likely to be chosen than another, did the system need to know who was on the ball for instance, if we pained over all the 'Carry On' stars faces with green paint, obscuring them, would Hattie Jacques be still more (or less) likely to be chosen than Kenneth Williams.

I eventually became victorious, proud in having brow beaten 20 heathens into logical submission, but all the time I was slightly saddened knowing that while I had spent the best part of a month playing with my balls the guy on the radio was probably going out, getting pissed, getting laid and enjoying himself.

[/weeps]

Feeling incredibly proud I read the whole thread - sadly not much the wiser......

#wishespaidmoreattentionatschool

The problem (as usually stated) is what are the chances of flipping a head 100 times in a row. Rather than in an arrangement of 100 coins what percentage of possible arrangements feature XXXXX (XXXX being whatever you are measuring).

Hey Help! I think I've been following most of this. Though straight up I should say I've no maths training. But whereas I can see Teome's reasoning I find it hard to see what you are getting at. Maybe you can clarify.

As Teome suggested, if you think of it as sequences, in a set of 100 restricted coin tosses the chances of ending up with 1 tail at some random point along the line and 99 heads is 100 times more likely (given that it has 100 different sequential possibilities than the single sequence possibility of all 100 heads). If this is correct (and unless I've misread, I think we're all agreed it is) then I dont see the relevance of your continuing argument about all sequences being *individually *equal in possibility.

Contrary to what you are claiming, I would say that Teome has phrased the argument in the manner in which we face it in everyday life: When we say that 100 heads or 100 tails is near impossible, we don't mean it is less likely as an individual sequence than 99 heads with specifically the second toss being a tail, or 99 heads with the 55th toss being a tail, etc. What we normally mean is that the chance of picking out that one random sequence is astronomically smaller than simply betting on ANY of the massive massive number of other undisclosed random sequences.

No? (Apologies if, despite my best efforts, I just sounded like a tellytubby)

Skully, the question you posed is related to the birthday paradox

we were actually discussing that at brockley drinks on friday which is why/when he posted this thread.

^^^^Yes but you're only considering a very specific example of equally probable events and the order matters. Others and myself were pointing out situations where the order is not important and often what is of interest is the frequency of occurance, not exactly when it happened. Probability Demsity Functions are based on the frequency of event divided by the total number of events and in physics we are often just interested in how many times. You're talking about the ways to arrange a system which is a part of it. I'm not arguing with your point but talking about more widely used probability.

If the runs were repeated infinitely many times all outcomes would occur. It's infinite! that's the basis of much of the theory, that for infinite runs all will occur

A typical example in physics is energy levels: two electrons e1 and e2 and two energy levels 0 and 1
There are four possibilities: e1_0 e2_0, e1_0 e2_1, e1_1 e2_0, e1_1 e2_1 but two give the same total energy. The expectation energy is
= 1/4 * 0 + 1/4 * 1 + 1/4 * 1 + 1/4 * 2 = 1
hence, the expected total energy is 1 and is the most probable outcome has probability 1/2 compared to 1/4 for 0 or 2 (arbitrary unitis). Much of classical statistical mechanics is based on the ways to arrange a system and calculations of the coefficients for a very simple 2 level quantum system. I/4 and 1/2 are the squares of the wavefunction coefficients. From the probabilities the entropy is calculated by

S = - sum [p_i * ln(p_i)] for all i

@eamonnog yes I agree. It's a different interest in the outcomes. All outcomes are equally likely but this tells us nothing useful and is not really worth analysing further unless you are picking lottery tickets, flipping coins or trying to disprove the randomness of a system. In many other cases the order and time of each event is less important and what is really of interest is the frequency of an event.

Just different cases and we are talking about probability in general. Once the probability of a system is know then one of the really interesting measures for physicists is the entropy,ie the disorder of a system,the inverse of how much is known.

Not arguing Help, just more broad applications

I find it hard to see what you are getting at. Maybe you can clarify.

By saying this:

"The problem (as usually stated) is what are the chances of flipping a head 100 times in a row. Rather than in an arrangement of 100 coins what percentage of possible arrangements feature XXXXX (XXXX being whatever you are measuring)"

I am saying that the problem as stated is asking for the probability of a set sequence (in this case a flip resulting in heads 100 times in a row) rather than the probability of the appearance of one of many sequences that share the ratios (H v T) of the original but not the order.

As Teome suggested, if you think of it as sequences, in a set of 100 restricted coin tosses the chances of ending up with 1 tail at some random point along the line and 99 heads is 100 times more likely (given that it has 100 different sequential possibilities than the single sequence possibility of all 100 heads).

It is a 100 times more likely because we are comparing 1 sequence to 100 sequences.

If this is correct (and unless I've misread, I think we're all agreed it is) then I dont see the relevance of your continuing argument about all sequences being *individually *equal in possibility.

Because they are. And because a sequence of 100 heads only has one order, there are no variations of 100 heads in a row.

Contrary to what you are claiming, I would say that Teome has phrased the argument in the manner in which we face it in everyday life: When we say that 100 heads or 100 tails is near impossible, we don't mean it is less likely as an individual sequence than 99 heads with specifically the second toss being a tail, or 99 heads with the 55th toss being a tail, etc. What we normally mean is that the chance of picking out that one random sequence is astronomically smaller than simply betting on ANY of the massive massive number of other undisclosed random sequences.

You have this part wrong.

The chances of that sequence (100 Hs in a row) occurring from a random selection is identical to ANY of the massive massive number of other sequences.

All of them are 1 in 1,267,650,600,228,229,000,000,000,000,000.

A hundred heads = 1 in 1,267,650,600,228,229,000,000,000,000,000
95 heads with 5 tails in the last 5 places = 1 in 1,267,650,600,228,229,000,000,000,000,000
46 heads and 54 tails bunched at either end = 1 in 1,267,650,600,228,229,000,000,000,000,000
HTHTHTHTHTHTHTHT . . . = 1 in 1,267,650,600,228,229,000,000,000,000,000
TTTHHHTTTHHHTTTHHH . . . = 1 in 1,267,650,600,228,229,000,000,000,000,000
One head at the start, one at the end and 98 tails in between = 1 in 1,267,650,600,228,229,000,000,000,000,000

The likelihood of ANY of the massive number of sequences (1.267 nonillion) occurring is identical.

You have this part wrong.

The chances of that sequence (100 Hs in a row) occurring from a random selection is identical to ANY of the massive massive number of other sequences.

All of them are 1 in 1,267,650,600,228,229,000,000,000,000,000.

The likelihood of ANY of the massive number of sequences (1.267 nonillion) occurring is identical.

We seem to be at crosswires yet again here. While I tried to be clear in my language, there was a small chance you could still take me up wrong and reply with the same answer... and that's exactly what you've done. But I'll take the blame and try again.

As I said, we are all absolutely agreed on the equal possibility of each individual sequence. The point is not really about maths at all at this stage. The disagreement is over the everyday meaning of the cliche about tossing a coin and getting 100 heads or 100 tails.

I'll give it one more go... while 100 tails or heads is of equal probability to any ONE other specifically noted sequence, this is not what people are betting about or generally referring to. When you bet someone that 100 head/tails is near impossible or that they couldn't possibily toss a coin 100 times and get 100 heads, what we are betting upon is: that ANY ONE among the astronomic number of different random equally possible other sequences is more likely to come up when you toss the coins, than any one specific sequence that they may choose.

It's like holding a raffle and giving them one ticket while you hold on to the rest of the book!

^^^^Yes but you're only considering a very specific example of equally probable events and the order matters.

Yep, agreed, a coin flip resulting in 100 heads is the target.

Comparing that to a series of other sequences ([99 heads and 1 tail] in one of 100 combinations) doesn't really work as a guide for probability as the original problem is searching for a specific sequence (1 sequence from 1.27n sequences) as 100 heads in a row has no variations even if we ignore the order of events.

The question as stated what is the probability of flipping 100 heads in a row? is looking for a single event not a group of events, so in that respect the order always matters, otherwise all we are saying is "hey look, these 100 sequences are 100 times more likely to occur that this 1 sequence, which is axiomatic.

Hope that makes sense !!!

If the runs were repeated infinitely many times all outcomes would occur. It's infinite! that's the basis of much of the theory, that for infinite runs all will occur.

I can't see that.

Let's keep the machinery light here - I have a perfect random number generator, switch it on and away it clicks, it produces either a 0 or a 1 every second, ignoring the cold death of the universe, a second term for the coalition or any other physical limitations (let's keep it all in an unchanging, hypothetical sempiternal universe) it keeps clicking away for an enormous amount of time, 0 after 0 after 0 after 0 after 0 after 0 after 0 after 0 after 0, nothing is broken, the machine is remotely checked without interference playing a part and all is well, it just happens to have only produces 0s so far. . .

By what mechanism would it be compelled to produce a 1 ?

S = - sum [p_i * ln(p_i)] for all i

'pips', I read 'pips', also 'sum', but I am going to stick with 'pips', how many points for 'pips' ?

@eamonnog yes I agree. It's a different interest in the outcomes. All outcomes are equally likely but this tells us nothing useful and is not really worth analysing further unless you are picking lottery tickets, flipping coins

100 coins in a row = order important (counterintuitively)

Lottery (6 numbers from 49) order not important.

• that ANY sequence at all* among the astronomic number of different random equally possible other sequences is more likely to come up when you toss the coins, than any one specific sequence that they may choose.
  
  

It's like holding a raffle and giving them one ticket while you hold on to the rest of the book!

Fixed that to make it even clearer what I was saying. : p

I'll give it one more go... while 100 tails or heads is of equal probability to any ONE other specifically noted sequence

Agreed, 1,27 nonillion sequences available from a 100 bit system (a hectobit) all have equal probability of occurring in a genuinely random system.

this is not what people are betting about or generally referring to.

Oh yes it is.

: )

When you bet someone that 100 head/tails is near impossible or that they couldn't possibily toss a coin 100 times and get 100 heads, what we are betting upon is: that ANY ONE among the astronomic number of different random equally possible other sequences is more likely to come up when you toss the coins, than any one specific sequence that they may choose.

This is where we disagree, it's simply not that case that (and I will quote what you say to avoid colouring your idea):

ANY ONE [I]among the astronomic number of different random equally possible other sequences is more likely to come up when you toss the coins, than any one specific sequence that they may choose.[/I]

Let's try an example, here is our 100 heads:

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

Can you type out (or cut and modify the above) a sequence from the other sequences more likely to occur ?

I guess what you are trying to say is that the chances of it (100Hs) not occurring is more likely than it occurring, this is true, but this equally true for all sequences.

Fixed that to make it even clearer what I was saying. : p

Ok, parsing, back in a second.

Yep, agreed, a coin flip resulting in 100 heads is the target.

Comparing that to a series of other sequences ([99 heads and 1 tail] in one of 100 combinations) doesn't really work as a guide for probability as the original problem is searching for a specific sequence (1 sequence from 1.27n sequences) as 100 heads in a row has no variations even if we ignore the order of events.

The question as stated what is the probability of flipping 100 heads in a row? is looking for a single event not a group of events, so in that respect the order always matters, otherwise all we are saying is "hey look, these 100 sequences are 100 times more likely to occur that this 1 sequence, which is axiomatic.

Hope that makes sense !!!

I can't see that.

Let's keep the machinery light here - I have a perfect random number generator, switch it on and away it clicks, it produces either a 0 or a 1 every second, ignoring the cold death of the universe, a second term for the coalition or any other physical limitations (let's keep it all in an unchanging, hypothetical sempiternal universe) it keeps clicking away for an enormous amount of time, 0 after 0 after 0 after 0 after 0 after 0 after 0 after 0 after 0, nothing is broken, the machine is remotely checked without interference playing a part and all is well, it just happens to have only produces 0s so far. . .

By what mechanism would it be compelled to produce a 1 ?

'pips', I read 'pips', also 'sum', but I am going to stick with 'pips', how many points for 'pips' ?

Makes sense alright! You're clearly too far up your own arse that you haven't considered that other people, including myself, were actually talking about both combinations and permuations. Maybe the question was phrased to be specific about this but you might want to consider that generally a conversation dips into other things.

Particularly when, as I said, the fact that all sequences are equally likely is trivial and tells us nothing because there is nothing that can be done with this. Hence, people were writing about combinations. You are welcome to keep banging on about all sequences being equally likely but this has been clear for quite a while now. Other aspects were being brought in and discussed.

As for pips, all I recognise that from is finance. Physics and the world is what I was getting at. Toodleoo

This is where we disagree, it's simply not that case that (and I will quote what you say to avoid colouring your idea):

ANY ONE [I]among the astronomic number of different random equally possible other sequences is more likely to come up when you toss the coins, than any one specific sequence that they may choose.[/I]

Oh come on this is getting ridiculous... I'd post a troll head here if I had the same forum posting skills as others on the forum. I specifically have said this is NOT what I'm saying... and I reposted precisely to try and stop you cutting my line to fit that interpretation. I've said over and over that each individual sequence is equally probable! And not only that but there's nothing complicated about that... we've all agreed!

Let's try an example, here is our 100 heads:

HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

Can you type out (or cut and modify the above) a sequence from the other sequences more likely to occur ? I guess what you are trying to say is that the chances of it (100Hs) not occurring is more likely than it occurring, this is true, but this equally true for all sequences.

You are getting closer to what I'm saying here... but it is not a pedantic point as you are making out... It is the only possible reasonable interpretation of the cliche about not being able to toss 100 heads or tails in a row!

Along with the raffle metaphor, the clearest I can make it is to say: that if I were to bet you that you couldn't toss 100 heads in a row, that bet is hardly dependent on me predicting the exact correct sequence that does show up!?!? I win if ANY, ANY ONE AT ALL, among the too-numerous-to-list alternative sequences comes up (i.e. if a tail turns up anywhere)!

The idea that the cliche about tossing a coin for 100 heads is dependent on someone else betting on an alternative equally probable specific sequence is absurd. Presumably this finally makes clear what me and teome were going on about. Your point is irrelevant except for being an interesting side-point.

You're clearly too far up your own arse . . .

You shouldn't let maths anger you, name calling doesn't clarify your position.

Oh come on this is getting ridiculous... I'd post a troll head here if I had the same forum posting skills as others on the forum. I specifically have said this is NOT what I'm saying... and I reposted precisely to try and stop you cutting my line to fit that interpretation.

I quoted from your own post (verbatim) - this was done before you edited your post and regardless of the edit I can't see how what I quoted from you was NOT what you were saying.

I am going to have to go with teome's 'up my own arse' and with your 'pedantic troll' and call it a day, we have obviously not convinced each other of our positions and as much as I am interested in this - and both your's and teome's posts - once the name calling kicks in these conversations start to lose their attraction for me.

[/flounce]

The idea that the cliche about tossing a coin for 100 heads is dependent on someone else betting on an alternative equally probable specific sequence is absurd.

Except for this, must reply to this.

[temporary de-flounce]

The probability of flipping a coin and producing a head 100 times in a row is not dependent on someone else betting on an alternative equally probable specific sequence, I am not sure anyone has made the claim that it is.

I think you are misunderstanding what I am saying, that - counterintuitively - the order of this exercise is important in working out the probability.

Is the order that lottery balls are drawn in a factor in calculating the probability of winning ?

No it is not a factor.

In a sequence of coin flips is the order of the 100 results a factor in calculating the probability of 'winning' (getting your 100H sequence) ?

Yes it is a factor.

The reason is that the lottery doesn't judge the order of the numbers (6,5,4,3,2,1 is identical to 1,2,3,4,5,6), but with coin flips the order matters because the probability of a specific sequence can only be measured against the absolute number of possible sequences.

So, whereas with the lottery 6,5,4,3,2,1 can be read as identical to 1,2,3,4,5,6 - with coin flips
H T H H H H H H (etc) is not identical to T H H H H H H H (etc) - if it were you would have much much less than the 1.27n outcomes.

Hopefully that has made what I mean a little clearer.

[/reflounce]

once the name calling kicks in these conversations start to lose their attraction for me.

I'll take it from here.

Knobheads.

When we say that 100 heads or 100 tails is near impossible, we don't mean it is less likely as an individual sequence than 99 heads with specifically the second toss being a tail, or 99 heads with the 55th toss being a tail, etc. What we normally mean is that the chance of picking out that one random sequence is astronomically smaller than simply betting on ANY of the massive massive number of other undisclosed random sequences.

^^^^Yes but you're only considering a very specific example of equally probable events and the order matters. Others and myself were pointing out situations where the order is not important and often what is of interest is the frequency of occurance, not exactly when it happened.

@eamonnog yes I agree. It's a different interest in the outcomes. All outcomes are equally likely but this tells us nothing useful...
Not arguing Help, just more broad applications

As I said, we are all absolutely agreed on the equal possibility of each individual sequence...

I'll give it one more go... while 100 tails or heads is of equal probability to any ONE other specifically noted sequence, this is not what people are betting about or generally referring to.

As you can see we've been posting the same thing all night. We all accepted that very simple point from the beginning but saw it as irrelevant to the main issue at hand. You simply chose to ignore these quotes and (rather infuriatingly) kept posting responses that made out that we didnt get it.

I quoted from your own post (verbatim) - this was done before you edited your post and regardless of the edit I can't see how what I quoted from you was NOT what you were saying.

I am going to have to go with teome's 'up my own arse' and with your 'pedantic troll' and call it a day, we have obviously not convinced each other of our positions and as much as I am interested in this - and both your's and teome's posts - once the name calling kicks in these conversations start to lose their attraction for me. [/flounce]

I fail to see how my post was in any way insulting. The troll face thing is a forum joke when you think someone is being amusingly provocative.

As I said in my last post... Your posts imply that the cliche about tossing a coin for 100 heads in a row is somehow dependent on someone else betting on an alternative (but equally probable) ***specific individual ***sequence. That is absurd. And I'm sure you'll find that out in a bar some night when you refuse to pay up coz the other fella didnt predict the exact sequence either ; )

saggy nuts

I used to wonder about the probability of winning the lottery on a lucky dip ticket, as you were asking the same numbers to be randomly selected in a 14m:1 chance twice.

I do however seem to recall that it has actually happened.

as a child? a lucky dip ticket has the same chance as any other. the lottery machine with the coloured balls off of the telly is not affected by the numbers on your ticket!