Probability - How does it work?!

There is a really good book on the history of Turing and the Polish mathmeticians efforts to crack Enigma by Khan which is worth a read for anyone into Cryptography. It's mostly descriptive so you don't need advanced maths to read it

Is the title or the author of the book located within your post but cunningly encrypted somehow so that working it out is an exercise for the reader? I only ask because I'd like to buy it but alas, the answer to the puzzle remains opaque :(

I was wondering the same thing.
I have this book at home but I'm not sure it's the one he is referring to

The Alan Turing: The Enigma: Amazon.co.uk: Andrew Hodges: Books

I have several others on hacking and code breaking, most of which detail the enigma cracking

Didn't mean to be so cryptic I just didn't have time to look it up. It's actually Kahn not khan. Rearangement error. I read it years ago and really recommend it. It's called The Codebreakers
http://en.m.wikipedia.org/wiki/David_Kahn_(writer)

A very good basic/mid-level overview of cryptography I read is
By Simon Singh - The Code Book

but hey don't let that stop you

Oh, I thought of quoting certain posts near the beginning of the thread but decided not to. I'm sure they will have been conscientiously edited by now. averts eyes :)

As I said probablity is completely relative
This is the book you need

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 think the first sentence is true, but the rest is false.

False bit first:
As has been established at tedious length no specific sequence is any more or less likely than any other for a fair coin. That's the definition of fair. (But as teome has elaborated, if you soften your focus and consider groups of possible sequences as 'the same', then you are more likely to get a sequence from a group that has more sequences in it).

The true bit:
In the real world we know that not all coins are fair. Some have two heads, and some have two tails, and some are biased in more subtle ways. Let's ignore all the possibilities apart from 'fair' and 'double heads' for now.

The rules of Bayesian probability tell us how to update our belief in each of these as we observe successive tosses, but first we need to decide what our initial belief is. Let's allow 0.0001 for 'double heads', leaving 0.9999 prior probability that this is a fair coin.

Now we start flipping the coin. The first toss comes up heads. This is consistent with both hypotheses but it should still cause us to adjust our belief in them. The double heads hypothesis did a better job of predicting this datum (it was certain we'd get a head, where as the fair coin hypothesis only gave that possibility 50%), so we should believe the 'double heads' hypothesis a little more than we did, and 'fair' a little less (by my rusty maths i get just under 0.0002 and just over 0.9998). After one flip we're still pretty sure the coin is fair, but if we go on getting continuous heads then at some point the fact that 'double heads' is doing a better job of predicting our data will overcome our initial dis-belief in it. With the prior probabilities i've assumed, after seeing the coin produce nothing but 14 heads i'd start to think it more likely than not that it isn't fair. (2^14 = 16384, enough to overcome our initial odds ratio of nearly 10,000)

Of course in the real world we could just pick up the coin turn it over and see whether the other side is a head too, but in the real world we should also allow more than two hypotheses that might explain the data. Maybe the person flipping the coin has the knack of making the coin spin and wobble without flipping so they can always catch it the same side up. Maybe when they let you examine it they switched their double headed coin with a fair one. Maybe someone's set up electromagnets that somehow control the way the coin lands when you flip it. Maybe this coin flipping you think you're doing isn't really happening - you're actually in a dream. Maybe space lizzards from beyond time are controlling all the air molecules to steer the coin... etc. You should probably give these possibilities far far less initial weight than the 'double heads' hypothesis, but as long as they keep doing a better job of predicting the data, you should keep increasing your belief in them and eventually start to take them seriously.

Help!'s position seems to be that the only hypothesis allowed is that the coin is fair. If that was really the case then he's right and all you can say on seeing 100 heads is 'ooh - that's pretty', but most people have enough prior knowledge and/or respect for their own ignorance to allow that something else might be happening.

If you have a fair coin there's nothing special about getting 100 heads from it, but if you've got 100 heads then there might be something special happening because there are so many other explanations for how you might have got them. Alternate explanations don't do as good a job of explaining 'random looking' sequences - either they have far lower prior probability (why on earth would anyone bother trick me into seeing this particular sequence?) or they spread their predictions over more sequences so don't perform that much better than the fair coin hypothesis.

...
Totally random numbers get the highest expected value from the Lotto, cause the other hipsters that decide to get totatally random number numbers get different random numbers.
...

I liked the rest of this post but i don't think this bit is necessarily right. For every number picking strategy there is an anti-strategy which consists of picking random sequences until you get one that wouldn't have been picked by someone using the strategy. If everyone else playing the lotto used the same strategy, you'd be better off with the anti-strategy than with just random numbers.

In general you need to double guess everyone else to try and avoid picking their numbers how ever you can. Of course if everyone picked according to the anti-strategy, you'd be better off with the strategy, but usually the anti-strategies allow a lot more different sequences than the strategies so are less likely to 'fill up'.

On the subject of cryptography, i think i read somewhere that CGHQ used the colossus computers they saved from the break-up of Bletchley park for filtering random sequences that were to be used in one-time pads. Taking out patches where the random sequence seemed to be following a predictable pattern technically made the pads less random, but leaving them in would have left patches where the pad didn't alter the clear text.

Yeah it's an extremely useful theorem. The false test results was the standard example in uni. I can't understand how it went unnoticed for so many years before being used in most areas of applied maths.

The fair coin calculation you did is a very nice use of it

if you're convinced your lottery numbers are going to win but are worried about someone else also getting it you should just play them twice.

this way if you win and have to share it with someone you get a larger portion of the pot.

if you have to share with one person you get 2/3 instead of 1/2
two people you get half instead of 1/3 and so on.

paradox is a massive misnoma in this case though! can't say i appreciate the headfuck aspect either tbh. clearly the former is 1/365 ignoring leap years, but the bigger the population in the latter case the more likely two people will share a birthday. i think even a mathematical shitwit would recognise that among 300 random people it is highly unlikely they all have different bdays

yeah, but what are the chances of getting sick on a saturday?

if you're convinced your lottery numbers are going to win but are worried about someone else also getting it you should just play them twice.

this way if you win and have to share it with someone you get a larger portion of the pot.

if you have to share with one person you get 2/3 instead of 1/2
two people you get half instead of 1/3 and so on.

Brilliant idea. I neve thought of that. Also, and this is something that I have really struggled to understand and still don't is this scenario.

You have 3 cards lying face down and on one of them is an X meaning win and you are asked to pick one card. If one of the two cards you did not choose is turned over reveals it to be blank, so you still have a chance of winning. If you are then offered the chance to swap your original choice for the other remaining card. You should swap as this increases your chance of winning. This is a headfuck and I think it is related to 'The Envelope Paradox'

the infinate monkey cage episode on probability and chance is great...

http://www.bbc.co.uk/podcasts/series/timc

My wife is a twin. Does that affect it? Is she more or less likely to have the same birthday as her twin brother?

If the chance of someone with a bomb getting onto the plane that you are travelling on is 1:100,000,000, do you reduce the odds by carrying a bomb yourself?

it does if they find it before you board.

probably

if you're convinced your lottery numbers are going to win but are worried about someone else also getting it you should just play them twice.

.

Been telling people that for years; people who buy five tickets with different numbers in order to 'increase' their chance of winning. Or not choosing 1,2,3 4, 5, 6 as mentioned above. Or numbers below 31.
Mostly I tell people not to take part in work lottery plays. Then, if you do win, you will be left with a very small share. That you collectively put in 50 tickets increases the chances of winning minutely. The counter-argument is that you don't want to see your colleagues win and not take a share yourself. Which is fair enough.
People also get trapped sticking to the numbers they have always used for fear that the one week they choose a new set, the old lot win.
I have never played the lottery. I would do, perhaps, but I would be too embarrassed to go in a shop and have to ask how to play it after all this time.

No embarrassment Will, just read this:
Lotto for dummies and northerners

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

I think it's in Taleb's book (Black Swan) the anecdote about a Maths type insisting that a coin that's landed 99 times on heads still has a 50% chance landing on tails the next throw, while some more streetwise trader suggests that he would place his mortgage on the coin landing heads again (because it's so obviously been tampered with).

Brilliant idea. I neve thought of that. Also, and this is something that I have really struggled to understand and still don't is this scenario.

You have 3 cards lying face down and on one of them is an X meaning win and you are asked to pick one card. If one of the two cards you did not choose is turned over reveals it to be blank, so you still have a chance of winning. If you are then offered the chance to swap your original choice for the other remaining card. You should swap as this increases your chance of winning. This is a headfuck and I think it is related to 'The Envelope Paradox'

This is also known as the Monty Hall problem. Easiest way to figure it out is to imagine there are 1000 cards. Imagine you picked one (1/1000 chance of getting the right one) and then 998 of the others were turned over. The person won't turn over the X card as he knows were it is, so there is now a 1/1000 chance your card is correct, and a 999/1000 chance that the other remaining card is; hence you should swap.

This is also known as the Monty Hall problem. Easiest way to figure it out is to imagine there are 1000 cards. Imagine you picked one (1/1000 chance of getting the right one) and then 998 of the others were turned over. The person won't turn over the X card as he knows were it is, so there is now a 1/1000 chance your card is correct, and a 999/1000 chance that the other remaining card is; hence you should swap.

Great way of explaining this one, simplest I have seen. Nice.

Been telling people that for years; people who buy five tickets with different numbers in order to 'increase' their chance of winning. Or not choosing 1,2,3 4, 5, 6 as mentioned above. Or numbers below 31.
Mostly I tell people not to take part in work lottery plays. Then, if you do win, you will be left with a very small share. That you collectively put in 50 tickets increases the chances of winning minutely. The counter-argument is that you don't want to see your colleagues win and not take a share yourself. Which is fair enough.
People also get trapped sticking to the numbers they have always used for fear that the one week they choose a new set, the old lot win.
I have never played the lottery. I would do, perhaps, but I would be too embarrassed to go in a shop and have to ask how to play it after all this time.

When the lottery first came out some enterprising contingency insurance underwriter tried to sell a policy to protect small businesses against losing their entire workforce in a syndicated win. That was until it was pointed out that the cheaper way of buying sufficient protection was probably to join the syndicate.

my uncle actually won the lottery as part of a syndicate. I got £250 in crisp £50 notes.

I spent most of it on a motorola pager. before most people had even mobiles... most notably any of my friends.

at age 16 I was a bit of a dick...

and now?

I'm a complete dick.