First, if you are in or around NYC, you can still take part in this great Halloween experiment into hauntings. Details here.
OK, onto the puzzle. Another classic from Martin Gardner this week.
Let’s suppose that a couple have four children. All four of the children could be the same sex, there could be three of one sex and one of another, or the sexes could be balanced two and two. Which possibility is most likely?
As ever, please do NOT post your answer, but do feel free to say if you think you have solved it and how long it took. Answer on Monday!
Advertisement
October 29, 2010 at 5:36 am |
I have a guess. It took me about two minutes to rationalize it.
October 29, 2010 at 5:40 am |
Got it.
October 29, 2010 at 5:50 am |
Had an initial gut reaction which, after thinking about it for two seconds, realized it was wrong. Thought about it, came up with an answer, now proving it, and yes, it is correct. Total time about two minutes.
October 29, 2010 at 6:00 am |
1½ minute to write it out
(assuming that the chance of having a boy is equal to having a girl [which in fact is not true; the odds of having a boy is slightly higher] and that there are no other sexes).
October 29, 2010 at 7:30 am
Why are the odds of having a boy higher? Or rather, where did you learn that?
October 29, 2010 at 8:54 am
Is it because many people stop having kids after they’ve got a boy? because in many cultures it’s seen as dishonorable not to have a boy, or something like that?
October 29, 2010 at 8:54 am
Roland is correct, in reality the odds of having a boy are ever so slightly higher than having a girl – more boys are born than girls. For the sake of problems like this though, it’s much simpler if you just assume a 50/50 boy/girl chance.
October 29, 2010 at 10:12 am
The probabilities pointed compensate a slight more child death toll in the boys than the girls, if I recall correctly…
Anyway, the answer to the puzzle is independent of the probabilities unless the probability of one sex is less than 20% (19.186% to be a little more precise), or to present it in another form, if there are more than four times of one gender birthing than the other…
October 31, 2010 at 7:32 am
It is a real phenomenon and I have heard the theory Joao Pedro Afonso is talking about.
To reply to Martha, no, that would not alter the ratio. That’s actually a common math problem that people often get wrong.
October 29, 2010 at 6:07 am |
Got it after 2 minutes, doing it the long way…
October 29, 2010 at 6:45 am |
First guess was confirmed by doing it the long way…now need to do the maths properly.
October 29, 2010 at 6:52 am |
about one and half and minutes.
October 29, 2010 at 7:01 am |
Bahhh!… this is a classical text book problem on probabilities…
Less than a millisecond, after reading the question mark…
(
When the Dragon’s puzzle will come?
October 29, 2010 at 7:29 am
That is impressive confidence. I think you should post another comment with your answer and workings on Sunday night, so you don’t spoil it for others but we can see if you got it.
October 29, 2010 at 7:54 am
I appreciate the vote of confidence, but seriously, don’t get your hopes up too much…
I didn’t pick the Best Puzzle Ever, and certainly not the Hardest Puzzle Ever. And although at the time I’d only ever seen one version of it, I’ve since learned that it is better known by another name, so true puzzle aficionados are likely to have seen it before in some form.
What I did pick was what I think is a pretty good puzzle (definitely better than average) that leaves room for different people to solve it in their own style. Also, it has a backstory in which someone gets killed if the puzzle is not solved, and you know how Richard loves those.
Part two of the version I sent involves proving that something is NOT possible, which is even more interesting. I don’t really expect Richard to include that part, and the official answer is actually erroneous, but I have quite a lot to say about it.
October 29, 2010 at 8:55 am
I have 2 coins totalling 25p. one of them is not a 5p. What are the two coins?
[Dragon] if it’s better than that one, it’s got my vote!
October 29, 2010 at 9:35 am
@Gareth, while I like to impress, my bad face has more to do with candy taste: Richard puzzles are funny and rarely I know them… so there is pleasure in solving them. But anyone who had any training/education in probabilities (and anyone with college education in technical areas, will usually had access to it in the first years surely), must have encountered this exercise in its first classrooms… it is almost standard. Richards description can only be consider a puzzle because people will likely forgot to sum… I’ll not say what at this moment, except to say that the ones with probability knowledge who’ll fail on it, didn’t read it carefully… that’s all.
But I don’t want to rant without giving something in return. So, here is another easy puzzle to the ones who didn’t like this one:
In a “backyard” country where couples has equal probabilities of having both sexes, there is a demographic law that says families can have one son per family… because heritage law are from father to son, and property can not be divided. So the way they do is, the couples have children but stop when they have one son… couples with no daughters, one daughter, two, three, ten, twenty daughters exists but only one son, always…
What you think? There is more boys or more girls in the country?
@Dragon, I’m not expecting it to be the perfect puzzle… it is just that, it should be surely be less textbook than this one.
October 29, 2010 at 10:24 am
20p and 5p…
October 29, 2010 at 4:05 pm
A missing word in my puzzle: can “only” have one son… sorry.
October 31, 2010 at 1:09 am
Are we allowed to answer Joao Pedro Afonso’s question here? I thought the math on that was interesting.
October 31, 2010 at 2:30 am
You can give any answers you want without spoiling anything for anyone by using tinypaste.com
Just write your answer or your spoiling comment, and paste the link here. ^_^
October 29, 2010 at 7:25 am |
Got it reasonably quickly using my probability knowledge, but then doubted my answer so wrote up all the possibilities and counted. Turned out I was right the first time.
There is something a little counterintuitive about which turns out to be most likely, but I think it’s right.
October 29, 2010 at 7:39 am |
Too many assumptions are used in these problems, ignoring the reality of how and why people choose to have another child after they already have one or more.
October 29, 2010 at 8:10 am |
A couple of minutes to write things out to check the answer.
October 29, 2010 at 8:23 am |
Why is this even considered a puzzle? It’s a straightforward calculation with no counter-intuitive or trick element.
October 29, 2010 at 10:55 am
I wouldn’t say the answer is particularly intuitive. I’ll try it on my wife later and see what she says – but, dare I say it, I doubt her “quick” answer will much resemble the actual one.
October 29, 2010 at 8:25 am |
My gut feeling turned out to be correct, so gut feeling – 5 seconds, writing it out, 2 mins. Double checking that I’d covered all bases with writing it out, another minute!!
October 29, 2010 at 8:31 am |
Took about a minute to calculate the probabilities
While it’s true births aren’t quite like unbiased coins the difference is only 1-2% which doesn’t change the solution.
October 29, 2010 at 8:38 am |
I did the same as the rest, it seems. A quick guess, a half-remembered puzzle from somewhere, then a scribbled check to prove myself right.
It is a little counter-intuitive, but easy to prove.
2 minutes?
October 29, 2010 at 8:44 am |
Got my theory quite quickly… and then totally disproved it in Excel. It makes sense when you see it on paper (or worksheet).
October 29, 2010 at 8:49 am |
The answer was quite easy if you consider probabilities (5 seconds to guess, 1 minute to write down permutations).
If you consider reality, it is getting even easier, even to guess the sequence, it is usually a different sex of the last child as parent try as long as possible to have children of both sexes. Just look around how many families with two older girls have a young boy or the other way round.
And it would be even easier if you were in China – you were only allowed more than one child if the first ones were girls (and if you were a farmer).
October 29, 2010 at 8:52 am |
Got it. Took a minute or two to write out the possible combinations.
October 29, 2010 at 8:52 am |
I’ve got a gut feeling about this, and since everyone else’s gut feelings seem to be right on the money, I’m going to call that a win!
October 29, 2010 at 9:00 am |
Binary!
October 29, 2010 at 9:23 am |
As worded, no distinction is made between two possible gender ratios – leading to one solution. If those two gender ratios are considered as distinct, that leads to another solution.
October 29, 2010 at 9:36 am |
Only last night I was reading a chapter on probability and the working for this very problem was described. What are the chances of that!?
October 29, 2010 at 10:16 am
Very high, conditioned to the chances of you have read a chapter on probability last high
October 29, 2010 at 10:13 am |
A couple of minutes for me too. I’m sure doing the maths would have been quicker and more impressive but I have a thing for colour-coded Excel tables.
October 29, 2010 at 11:32 am |
Just a few seconds, but no idea if I am right or not yet.
October 29, 2010 at 11:41 am |
Got it instantly. verified in 20 seconds
October 29, 2010 at 11:56 am |
About 45 seconds to write out the possibilities. It’s not the answer I thought it would be if I’m right…
October 29, 2010 at 12:05 pm |
It’s taken me three days so far.
October 29, 2010 at 12:27 pm |
I knew it immediately, but that’s because I play Bridge.
October 29, 2010 at 12:31 pm |
Thought I had it instantly. Worked it out for a minute, and confirmed my guess. Then I read the comments, and realized that I had misread the puzzle and had the wrong answer. So about 2-2.5 minutes all together.
October 29, 2010 at 1:11 pm |
Not exactly a puzzle, it’s just genetics
October 29, 2010 at 1:28 pm |
Hehe the “disttibution” of my kids the most likely one, even though not so surprising..
October 29, 2010 at 10:47 pm
that’s also the distribuition of my dad’s kids, : )
I’m a child of statistics
October 29, 2010 at 1:37 pm |
I think I arrived at the correct answer after a minute or so. I’ll wait until Monday for confirmation! Intersesting probability type puzzle that.
October 29, 2010 at 2:11 pm |
I believe I have the answer, and it took about 30 seconds.
October 29, 2010 at 2:32 pm |
Actually, I’ve revised my answer after thinking about it some more. I’m certain I have the right answer now.
October 29, 2010 at 2:55 pm |
Listing the possible combinations was the quickest way. Then bunching them according to the wording of the question. If I get it wrong, then my grasp of probabilities is seriously flawed.
October 29, 2010 at 3:11 pm |
Since this is a Martin Gardner puzzle, there must be more here than meets the eye, i.e. independent binomial trials, but I’m not sure what it is.
October 29, 2010 at 3:49 pm |
[...] It's the Friday Puzzle! « Richard Wiseman's Blog [...]
October 29, 2010 at 4:00 pm |
Ah, Yes. Mark is right. The important thing is the wording of the problem.
October 29, 2010 at 4:02 pm |
As a bridge player of moderate competence, I know the answer to this kind of thing instantly. Anyone who has any instinct for that kind of card game will know this quite intuitively.
The maths used here is GSCE level, so really everyone in Britain ought to have studied it in class. But probably a lot of people weren’t listening, or were smoking behind the bike shed, or something.
October 29, 2010 at 4:44 pm |
My gut feeling was tossing up between the obvious answer and the less obvious one.
After 15 seconds of thinking, I rationalized it to be the obvious one.
A minute later, I rethought and chose the less obvious one.
Okay, I’m sticking with this one.
October 29, 2010 at 6:37 pm |
Took about 2 minutes to write it out as a chart and then count the answers. I was a little surprised at the result, but it seems right.
October 29, 2010 at 6:53 pm |
Pretty sure I got it, but I’ve got this feeling this is going to be one of those things that gets argued about endlessly…
October 29, 2010 at 11:00 pm |
I’m lying in bed, so worked out using four fingers.
October 30, 2010 at 12:06 am |
If you’ve ever taken a course on statistical mechanics, you’d immediately know the answer.
October 30, 2010 at 5:43 am |
The answer is dependent on the phrasing of the question.
If there are only three possible groups and no specific gender differentiating ratios:
1. All four of the children could be the same sex
2. There could be three of one sex and one of another
3. The sexes could be balanced two and two.
Then one ends up with solution X
If the question is phrased in such a way as to incorporate gender distinguishing groups:
1. 4 males and 0 females
2. 3 males and 1 female
3. 2 males and 2 females
4. 1 male and 3 females
5. 0 males and 4 females
Then an alternate answer to ‘solution X’ will be obtained.
October 30, 2010 at 12:00 pm |
Having been stung by this kind of probability question before, I spent a minute with a calculator and then realized the trick.
October 30, 2010 at 12:19 pm |
2min! but with Probability theory!
October 30, 2010 at 2:32 pm |
I had an idea for one answer, but then realised there was no set order required so came up with another.
October 30, 2010 at 5:53 pm |
Spent a couple of minutes drawing it out and I think I got it right. I’ve no memory of the math needed for this (My last mathematics class was 20 years ago), but I’m counting on logic and pen and paper to see me through, not to mention what I’ve learned from past puzzles on this site.
October 30, 2010 at 9:45 pm |
Isn’t the answer based more on modern lifestyles, rather than actual math and/or statistics?? If set in 1850, the answer would be quite different, I beleive.
October 31, 2010 at 10:10 pm |
30 seconds, using brute force (and finger binary tapping). I found it interesting. I didn’t know it, even though it sounds quite basic.
Solution:
http://tinyurl.com/2a4vxvn
November 1, 2010 at 4:50 pm |
Well, since it had been mentioned it didn’t take much!!