Probability - How does it work?!

was it the ipod that when on random sometimes played the same song twice in a row, as should be expected if the songs are selected at random. The music selection software was changed after complaints to make it "more random"

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.

You mean once when they are selected in the shop and then once again when they are selected in the actual draw ?

I think you are getting some stuff wrong here, the odds of getting those initial random numbers have no bearing on the odds of those numbers being selected in the draw.

So it is still 1 in 14m - rather than 1 in 14m x 14m.

What about those born on the 29th of Feb?

Those numbers assume a 365 day year.

Oh, I've just noticed. I thought it's 356.

Sharing the winnings with all the other clever clogs who picked 1-6 would be a bit annoying though? ;)

31 FTW apparently (avoids birth date pickers)

Yep, all that stuff should be taken into account, but that is 'outside' of the maths, once we know the maths we can factor in things like buying the ticket as late in the week as possible (as you have more chance of dying if you buy the ticket earlier).

was it the ipod that when on random sometimes played the same song twice in a row, as should be expected if the songs are selected at random. The music selection software was changed after complaints to make it "more random"

It's 'pseudorandom' rather than 'random'.

I would guess, from what you wrote, that Apple made the selection less 'random' to get around this, they would have stuck a stage in the PRNG (pseudorandom number generator) that simply checked the output for a duplicate of the previous output and if found got it to try again.

sounds like you can 'do the math'

Yep, all that stuff should be taken into account, but that is 'outside' of the maths, once we know the maths we can stop doing the fucking lottery

xD

xD

Yep, it's an idiot tax.

The probability of 100 heads in a row is identical to any other combination.

But surely in the case of only 100 tosses (fnarr) 100 heads in a row would mean a coin that'd been tampered with? Shouldn't it even itself out to 50/50 over the full 100? Whereby 50 heads in a row is equally likely as 50 tails in a row, or heads-tails-heads-tails-heads-tails etc or any other combo as you say?

I've just finished reading a book about exactly this topic ( The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern Basic Ideas: Amazon.co.uk: Keith Devlin: Books ) and it seems that even after reading it I am still having trouble grasping the ramifications of this stuff.

For further trouble grasping ramification and for a fascinating exploration of mind maths and creation this is amazing:

Is there a third 'flip a coin' outcome, of the coin landing on it's edge?

"Box number 7 please Maureen"

For further trouble grasping ramification and for a fascinating exploration of mind maths and creation this is amazing:

Hrm. I read his second book, 'I am a strange loop' and I loathed it. It always seemed barely one step away from crystal bowls and chanting, and he'd devoted page after page to calling Bertrand Russell a coward for attempting to define basic axioms about mathematics(!?) Apparently he wrote strange loop because people didn't 'get' GEB and he wanted to make it more explicit what he was trying to say. The main things that I took away from it were that if you didn't share his world view then you, too, are a coward like Russell, and not only a coward but stupid too - just like the people who read GEB and didn't 'get' what he was trying to say with it. The problem was that his world view seemed to be using Godel's incompleteness theorem to 'prove' that his dead wife lived on in his own consciousness as some kind of metaphysical recursive function. Too bonkers for me. I know he's generally well respected but strange loop really put me off his work.

Penrose's Emperor's New Mind dealt with similar themes (and does in fact address Hofstadter's work directly, sans dead wife) but did it much more logically and without any of the supporting woo:
The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics Popular Science: Amazon.co.uk: Sir Roger Penrose: Books

Shouldn't it even itself out to 50/50 over the full 100? Whereby 50 heads in a row is equally likely as 50 tails in a row, or heads-tails-heads-tails-heads-tails etc or any other combo as you say?

Over 100 coin tosses, there are 2^100 possible distinct sequences, and they're all equally likely (ie, 1 in 2^100 chance of each).
However, some outcomes can be reached by more than one sequence, so they're more likely to occur.
For example, there is only one way to get either all 100 heads or all tails, so those outcomes are vanishingly unlikely (2^100 has something like 30 zeroes).
However, take the outcome 1 head and 99 tails ... there are 100 sequences that give you the outcome, so it's 100 times more likely to come up.
The outcome 50 heads and 50 tails can be accomplished so many ways, it's much more likely to occur: look up 100 choose 50.

Easiest way to work through this is to start with small numbers (<10) and write out the tree.

Didn't read strange loop and probably (66.66"%) won't after your critique bothwell.
I really did enjoy GEB. It was entertaining to read esp the carolesque interludes
and the way he illustrated the mathematical points using exambles from art and music
and the way he illustrated the musical points using examples from maths and art so
I really didn't enjoy not reading BEG. It wasn;t entertaining not to read especially the recursive escher interludes
Will probably (33.33%) read strange loop thanks to your critique bothwell

That Penrose book is quantum mystic crap.

Didn't read strange loop and probably (66.66"%) won't after your critique bothwell.
I really did enjoy GEB. It was entertaining to read esp the carolesque interludes
and the way he illustrated the mathematical points using exambles from art and music
and the way he illustrated the musical points using examples from maths and art so
I really didn't enjoy not reading BEG. It wasn;t entertaining not to read especially the recursive escher interludes
Will probably (33.33%) read strange loop thanks to your critique bothwell

You've missed out the probability of reading half of it before being struck by a meteor. Back to the probability tree with you!

That Penrose book is quantum mystic crap.

Ha! I enjoyed it. I've no opinion either way on his final conclusion (that consciousness itself is somehow related to quantum gravity) because I couldn't really see how he got to it, but I do agree with his misgivings over consciousness being algorithmic and I enjoyed the highly ambitious scope of his conjecture.

Shouldn't it even itself out to 50/50 over the full 100?

No. On average it should, but each toss is an individual event, in no way related to the previous one (as long as you assume there is no damage to the coin, surface, etc), so even if the previous 99 where heads, it makes no difference to the 100th throw.

Yeah. I think I'm kind of understanding after useless's very useful post #65. At least until the problem gets rephrased into a different scenario and my tiny brain can't fit it together again :(

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

That Penrose book is quantum mystic crap.

Yeah he's been criticised for this view by many people. He's a great physicist but there is a lot of guess work going on there. The idea that consciousness is a quantum mechanical phenomenon is very interesting. Christof Koch has done a lot if work in the area and dismisses the quantum effects but its not a supprise that traditional neuroscientists would
Not like the idea since they haven't got a clue about QM.

But surely in the case of only 100 tosses (fnarr) 100 heads in a row would mean a coin that'd been tampered with?

No need for coin tampering, in fact no need for coins at all, as a purely mental exercise you can see how a heads > heads > heads > heads > heads > heads > heads > (etc) outcome is as likely as any other.

Shouldn't it even itself out to 50/50 over the full 100?

No, there is nothing pushing the system to 'even itself out'.

Whereby 50 heads in a row is equally likely as 50 tails in a row, or heads-tails-heads-tails-heads-tails etc or any other combo as you say?

Yep, all combinations are equally likely.

Over 100 coin tosses, there are 2^100 possible distinct sequences, and they're all equally likely (ie, 1 in 2^100 chance of each).
However, some outcomes can be reached by more than one sequence, so they're more likely to occur.
For example, there is only one way to get either all 100 heads or all tails, so those outcomes are vanishingly unlikely (2^100 has something like 30 zeroes).

All the outcomes are vanishingly unlikely, a row of 100 heads no more or less that any other outcome.

Chances of heads 100 times in a row 1 in 1,267,650,600,228,229,000,000,000,000,000
Chances of any sequence in 100 flips 1 in 1,267,650,600,228,229,000,000,000,000,000

However, take the outcome 1 head and 99 tails ... there are 100 sequences that give you the outcome, so it's 100 times more likely to come up.
The outcome 50 heads and 50 tails can be accomplished so many ways, it's much more likely to occur: look up 100 choose 50.

But, each of these sequences is individual and as likely to come up as any other outcome. Think it as a 100 bit binary system where each coin is a row of 100 coins needs to be set to 0 or 1 (heads or tails).

100 bit binary = 1,267,650,600,228,229,000,000,000,000,000 variations (1.267 nonillion).

^yes, each specific sequence is equally likely but it depends on whether the order has to be considered. 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. This is the way it 'evens' out when order doesn't matter and after many many tosses there will be a bell shaped curve with equal number of heads and tails being the highest frequency outcome and only heads or tails being the two lowest.

If a run of 'x' coin tosses was repeated infinitely many times each possible sequence would occur but when counting how many heads or tails occurred in each run the condition of equally many heads and tails is the most common --> probable.

Chances of heads 100 times in a row 1 in 1,267,650,600,228,229,000,000,000,000,000
Chances of any sequence in 100 flips 1 in 1,267,650,600,228,229,000,000,000,000,000

Yes but if heads came up 100 times in a row the chance of me noticing and shouting this from the rooftops 100% , whereas for most other combinations I probably wouldn't.

(though the chance of me tossing a coin 100 times to see what happens 2%)