Quote NorthernBulls="NorthernBulls"2 and 4 are exactly the same result you are trying to say 10 + 5 = 15 but 5 + 10 doesn't = 15.
So only two possibilities which are equally likely so it's a 1 in 2 chance of being a boy.'"
No, i have set it out very clearly. this is the problem with people failing to understand probability. Even when you set it out, if they find it counter-intuitive, many people still won't have it. Don't worry - it's a common trait.
But clearly, having a girl first, and a boy second, is not the same thing as having a boy first, then a girl. This doesn't need explaining.
Quote NorthernBulls="vbfg"Why are the permutations of their birth order significant? There's one child, and there's the other one. '"
Because on the information you have, you do not know whether Myrtle is the older or younger child.
Now, if someone gave you additional information, and revealed that Myrtle WAS (say) the OLDER of the two children, that alters the probability. Why? Because now, you can exclude the Boy-Girl order of birth. That leaves 2 remaining possibilities, Girl-Girl, or Girl- Boy. And is an even money shout.
But if, as in the original scenario, you don't know whether Myrtle is the elder or younger, then Boy - Girl remains an equal possibility. That is the significance.
Statistically, if you picked a random sample of 100 families where they had had two children, and in each family at least one child was known to be a girl, it follows that on average you would expect to find an even distribution of birth orders. That is, there should on average be 25 G+G, 25 G+B, 25 B+G. There can't be any B+B. So that leaves 75 families standing.
If you have a £ on each family, and bet that in each case, the other child is a boy, then you will lose £25 on the G+G families, but you will win £25 + £25 = £50 on the G+B and the B+G families.
This is why the person who wrongly thinks, that in the given case of any of the families, it is an even money bet, can lose a lot of money. Exactly this principle is used to tempt mugs the world over in an extremely common three-card trick treet scam.