So here goes my first post of 2013 (and I'll spare you the mathematical factoids about the number 2013: my students seemed to know them all from Facebook before I told them, anyway).
A few days ago a friend with whom I did my PGCE posted a really rather excellent maths problem on Facebook, claiming it to be better than Monty Hall. It went like this:
I have two children. One of them is a boy born on Tuesday. What is the probability I have two sons?I managed to work out the answer within twenty minutes (not bragging, just sayin') but I kept thinking about it for several hours afterwards, and then spent all week talking about it to anyone who would listen, and even those who wouldn't (sorry colleagues, sorry boyfriend, sorry dad...).
But that's not actually the problem I'm going to discuss. As interesting as I found it, I found that the people I discussed it with got themselves hung up on a less interesting part of the puzzle. I couldn't believe that something I thought was so clear cut was causing so many people such cognitive conflict.
So I am going to discuss the Two Children Problem (the problem above but without the Tuesday bit).
I thought it would be fun to lay this out in a Sophie's World kind of format. I am going to discuss the problem via the emails sent between my dad and myself for the past few days. These are almost entirely unedited. Enjoy!
Me: I have a good probability puzzle for you. I have two children. One of them is a boy. What is the probability I have two boys?
Dad: Off the top of my head, I would guess 50 – 50.
Me: It's not.
Dad: I don’t understand. The other child is either a boy or a girl. That means there’s a 50% chance of it being a boy. What’s the snow like in Coventry is it bad?
Me: There are four options: BB, BG, GB, GG.
I have eliminated GG by telling
you one is a boy.
So the probability would be
1/3.
It is a bit like the Monty Hall
problem (two goats, one car).
The snow is mental!! Got sent
home from school at 12:30. Walked home. In un-walked areas it was about 15cm
deep.
Dad: No there are only two options because:
GG is not possible and GB and BG are the same option
(nothing is mentioned here about whether one is older or younger). Hence the
only options are BG or BB.
Me: But they are not the same. I didn't point at a child and say this one is a boy.
I said one of them is a boy.
Dad: Yes and so if one of them is a boy, the other is either a boy or a girl.
Me: No!
Flip two coins. If both are
tails, re-flip them both. If not, then one of them is heads. What is the chance
the other one is also heads?
Dad: I have to go home now. Send a more explicit
explanation to my home e-mail. I still don’t believe
you.
Me: f you flip two coins, and reflip them if they're both tails, then at least one of your coins is heads. What is more likely, they're the same or they're different?
Dad: I don't understand this analogy with coins. Surely the analogy is that:
You flip one coin and it turms out tails (a boy). That is now a known. If you then flip another coin it may be heads ( a girl) or tails (a boy). So the question is if you flip one coin and it turns out to be tails, and then you flip another coin what are the chances of it being the same or different as the first coin? With the second coin the probability is 50 - 50 of it being either heads or tails.
Is this not the case?
Me: This is not the same. You are right that if you flip one coin and it is heads, then the other coin is 50-50. But what we have done is flipped both coins, looked at both of them, and noticed that at least one of them is a head. So there are three possibilities: HH, HT, TH.
Dad: All right, I think I get it now.
If you flip 2 coins at the same time, what are the chances of them both being heads, if one of them is a head. Then the answer is 1 in 3.
H T
T H
H H
Therefore
B G
G B
B B
I think that what troubled me here was a notion of sequence. I am still thinking about this.
What about this? There are 2 goals in a football match. Palace have scored 1 of the goals. What is the probability that Palace have scored both goals?
What is the probability of Palace scoring both goals? Surely the score is either
Palace1 : 1 Bolton or
Palace 2 : 0 Bolton
Therefore there is a 50 - 50 probability that Palace have scored both goals, (and in the real world Glenn Murray probably scored both of them).
What is the fallacy if,
Goals equals children
Palace goals equals boys
Therefore the score is either
1 boy : 1 girl
2 boys : 0 girls
Palace1 : 1 Bolton or
Palace 2 : 0 Bolton
Therefore there is a 50 - 50 probability that Palace have scored both goals, (and in the real world Glenn Murray probably scored both of them).
What is the fallacy if,
Goals equals children
Palace goals equals boys
Therefore the score is either
1 boy : 1 girl
2 boys : 0 girls
Me: Hmmm this is a really interesting question. I am thinking about it.
Dad: What about this. If you put two blue counters (boys) and two red counters (girls) in a bag. If you take out a blue counter, it immediately negates the existence of one of the red counters, because it is no longer a possibility. Therefore for next pick there is one red and one blue counter in the bag and therefore a 50% chance of it being blue.
With the coins, if for example, the first coin is a H then it negates the possibility of
T H
therefore there are only two possibilities;
H T or H H.
If the second coin is a H, it similarly negates the possibility of H T and so there are only two possibilities T H or H H. Therefore in both cases if you know that at least one of them is a H, there is a 50% chance that the other is also H.
Me: You are correct in what you have said above, but that is a completely different situation.
The original problem can be thought of like this:
Pick a million families that have two children. Discard all those that have two girls.
Are you really saying that half of these families will have two boys, and the other half will have one of each?
Surely you can see that that is not true in real life.
As for the football question, let's think of it in the same way:
Pick a hundred Palace matches where two goals have been scored, eliminating all those where Palace did not score.
Of these matches, do you think Palace will have won half and drawn half, or do you think they are more likely to draw?
Dad: Let's say 990,000 families instead of 1,000,000. Discard families with two girls. Is that 330,000? That leaves 660,000. Half of those are families with two boys. That is 330,000. Wouldn't that fit with real life? You end end up with 50% boys and 50% girls. (Actually in the real world I think there are slightly more female births than male births).
Me: You are wrong. I think it's more like this:
1 000 000 families.
250 000 two girls
500 000 mixture
250 000 two boys.
If you don't agree, consider a family with 4 children.
Combos are:
0 boys 4 girls
1 boy 3 girls
2 boys 2 girls
3 boys 1 girl
4 boys 0 girls.
You might think these 5 are equally likely. But actually they're not.
The 2 of each option is actually 3 times more likely than having 4 of one.
This is based on the Binomial theorem. Are you familiar with that? Do you know what Pascal's triangle is?
Dad: If you flip a coin and it turns out H does it make it more likely that when you flip another coin, it turns T? If so, why?
Yes I can see that the combinations are
B B
G B
B G
G G
And therefore a 25 percent distribution. But again if each birth invoves a 50 - 50 chance, how does one birth apparently influence another?
B B
G B
B G
G G
And therefore a 25 percent distribution. But again if each birth invoves a 50 - 50 chance, how does one birth apparently influence another?
Is pascal's triangle a kind of musical instrument?
Me: One birth does not influence the other. But knowing that one of them is boy only tells you that GG is not an option. There are still three equally likely options left. Note that by "one of them" what we really mean is "at least one of them".
Dad: Interestingly in 40 football matches where there were at least two goals, 23 of which were 1- 1 after 2 goals and 17 were 2-0 or 0-2. Therefore approx 58 perc to 42.
Me: Well that supports my argument! Convinced yet?
The conversation ends there (for now). Do you think the answer is a third or a half? I find it weird that my dad still doesn't quite believe me. I suppose that is the difference between a mathematician (me) and a linguist (him). I look for meaning by extracting the bare minimum of information to simplify the problem, whereas he looks deeper in between the lines to find meaning.
It goes without saying that my way of thinking is better. :-)
Emma x x x
PS Look out for part two, where I'll discuss the Tuesday boy problem.
