Bramble said:
The odds of being hit crossing the road are not 50/50. Unless you're stupid.
And the odds of flipping a head is not dependent on any previous run of tails where each result is viewed as a 'one off 'event.
However...viewed as a continuum of events...the chances on NOT flipping a head after N flips of a tails does decrease when viewed as a series of possibilities.
I was wrong to apply a coin flip analogy to the road crossing as we are not dealing with a mathematical model with fixed variables. Just because sulong's hero was hit on the last crossing after 14599 successful ones does not imply a probability of 1/14600 of being hit in the future unless we create a bizarre arbitrary model. I tried doing that and then ran into further problems, such as "Does the model
ensure an accident occurs every 14600 crossings, in which case with each crossing the probability of being hit
will increase, until it eventually becomes 1if our hero has miraculously survived 14599 crossings, or does it behave like a 14600-sided die in which case the probability will stay the same regardless of the past."
Actually you might disagree with that last statement so let's return to coin flips for simplicity. You say there is a difference between viewing a series of flips as a continuum and viewing the flips as one-off discrete events.
I'm pretty sure I disasgree with that, though I'm finding it hard to explain exactly why because my intuiton almost agrees with you! We have to be very careful with definitions in order to get anywhere useful ...
Let our experiment always consist of ten coin flips and try and see what happens when we view these flips both with hindsight and 'blindly' during the sequence.
If we predetermine exactly what sequence we require then we can determine the exact probability of that happening. For instance, the chance of flipping nine tails then a head is exactly the same a flipping ten tails, or three heads then seven tails, or indeed any fixed predetermined sequence. In each case it is 1/1024.
Assume a run of nine tails. Pretty unlikely. A chance of 1/512 in fact. Now, given this run, is the next flip equally likely to be a head or a tail? Yes, it is. Nine tails then a head has a chance of 1/1024 as does nine tails then a tail.
I don't think we have the luxury of viewing the sequence as a continuum, if that makes any sense. Aaargh I'm going round in circles.
However...viewed as a continuum of events...the chances on NOT flipping a head after N flips of a tails does decrease when viewed as a series of possibilities.
This is the same as saying "However ... viewed as a continuum of events ... the chances of flipping a head after N flips of tails does increase when viewed as a series of possibilities."
Granted, the probability of N flips of tails in an unbroken future sequence decreases as N increases, but the crucial point here is that N flips of tails
has already happened! It is in the past - you said so by inserting "after". So the chances of then flipping a head given the past sequence of tails does not increase, it stays the same, as does the chance of flipping another tail. "When viewed as a series of possibilites", given a past sequence X, the chance of flipping a head or a tail stays the same.
This post has given me terrible déja vu and I'm still not entirely happy with it
Strange old thing probability! Anyway, I'll leave it there before I give everyone a headache. I suspect we are arguing semantics more than anything more fundamental.
An aside to this is the matter of coincidence. How often do people say "How amazing! I bumped into someone in the supermarket today with exactly the same name as me, and her daughter goes to my old school and do you know she's even wearing exactly the same dress that I bought last month from a boutique in Nimes! What are the chances of that!"
Well, the chances of someone meeting these predetermined conditions would be minuscule, granted, but of course she didn't predetermine them. It is likely that there would be an immense pool of things the two people might have in common and the three mentioned represent only a fraction. In fact it would be unusual for coincidences like this NOT to happen as there are so many ways in which they can. Put another way, some unlikely event is likely to occur, whereas it's much less likely that a particular one will. If you don't specify a predicted event precisely, there are an indeterminate number of ways for an
event of that general kind to take place. The paradoxical conclusion is that is would be very unlikely for unlikely events not to occur. (with thanks to Paulos for that).
