Analogy I was given and roughly what was used on here.
If I tossed the coins, and hid them from you (one in each hand), and
said “at least one is a head”, then you would say “OK, so I know that
one is a head, doesn’t matter which one – I’ll assume it’s the one in
his left hand. So all that I don’t know is what’s in his right hand –
50/50 chance. Easy.”. But that’s not right. It’s not obviously wrong,
but it *is* wrong. You could analyse it another way:
“What’s in his left hand? Could be a head or a tail – I don’t know.
Suppose it’s a head, then there is an equal chance that his right hand
contains a head or a tail. So – *assuming* his left hand has a head,
then there we have two equally likely outcomes. Suppose his left hand
actually has a tail. Then his right hand *must* contain a head because
we know there’s at least one head. There’s a third outcome. Since the
coins were tossed fairly and randomly assigned to each hand, those
three outcomes are equally likely, and we’re back to our 1/3, 2/3
probability.