Originally posted by wolfgang59
I toss two coins, lets say a £1 coin and a £2 coin they land unseen by me.
A friend tells me that I have tossed "at least one Head"
What are the chances that the coins are different? (ie H & T)
My friend's information tells me that the result is one of these three.
£1 head and £2 head
£1 head and £2 tail
£1 tail and £2 head
The c ...[text shortened]... l the time) my chances of tossing a head and a tail are 2/3.
But we all know its 1/2.
let's say you specifically asked
your friend to watch the coins and say whether or not there was at least one head.
then these are the scenarios:
HH - "yes, at least one is a head"
HT - "yes, at least one is a head"
TH - "yes, at least one is a head"
TT - "no, neither one is a head"
in this instance, you have clearly eliminated the TT possibility, and the chance of mixed coins is 2/3.
You could have also asked your friend to tell you whether or not there was a tail - but in both cases, the friend could only give info about the coin side you asked him about
now - we look at the scenario that actually occurred - we will assume that your friend would have given some sort of a response - and if the coins are mixed, that he would have been equally likely
to talk about heads or to talk about tails
suppose we made 40 flips with each possibility happening an equal number of times
HH - "there is at least one head" (10)
HT - "there is at least one head" (5) and "there is at least one tail" (5)
TH - "there is at least one head" (5) and "there is at least one tail" (5)
TT - "there is at least one tail" (10)
you hear "there is at least one head" - this statement would occur 20 times during the 40 flips. Of those 20 times, 10 would have occurred after HH, while only 5 would have occurred after HT and 5 would have occurred after TH.
As a result, half the time you heard this statement, you'd have two heads, and half the time you'd have mixed coins
the key to the paradox, is that when the coins are the same, the friend is forced to mention that coin side, but when the coins are mixed, your friend has a choice of which coin side to mention