November 27, 2012

In one hospital, over a period of 14 hours, 19 boys — and no girls — were born.

It's in the news as if it's amazing, but of course it's no more unusual than any other combination of boys and girls born anywhere else over any other period you might define. In fact, it's less unusual, since the choice to define the relevant period and end it when the last boy was born maximized the presence of boys in the group.

89 comments:

Chip S. said...

What's amazing is that they've all received solicitations to contribute $3 to Organizing for America already.

elkh1 said...

Any pre-natal scans? Any aborted girls? We don't know, do we? Are we on the way to China?

chickelit said...

Did Nate Silver give odds on that probability?

Joe P. said...

Your intuition that it's not news is completely on track.

If you assume total randomness and 50/50 odds between boys and girls, the odds are about one in 8000 that this happens starting with any specific birth (and assuming you don't care about boys-in-a-row vs. girls). That is any birth, no matter how many of the same sex came just before it at the same hospital.

Given the 4m births in the US this year, assuming they were all in hospitals, give or take, this event should have happened an average of 500 times, among all possible boy-girl birth combinations.

So why did they publish? Maybe because they were bored? It's local news.

Rob said...

Most reporters, not to mention most lawyers, are incapable of calculating the odds of a sequence such as this.

MrCharlie2 said...

yes, not news, just struck the people doing the deliveries that it was boy after boy for 3 days.

In China perhaps it wouldn't seem strange

Bob Ellison said...

Seems newsworthy to me.

I once opened a bag of Skittles and poured a sample into my hand. All five came out purple. I mentioned this to my golfing partners, who were both software engineers, and suggested that the probability of getting five of the same color at first open from a bag with five colors was pretty small. They didn't take the bait. They continued golfing, badly.

The rest of the bag did, indeed, contain other colors, and the bag had been jostling about in my pocket for at least a day or two before that, so no answer comes from assuming packaging problems.

Well, I guess you hadda be there. A one in eight thousand thing happens once in a while, and it's interesting when it does. That's part of why I bought a few tickets in the current Powerball lottery.

BarrySanders20 said...

If you started randomly and assume there are only two choices (male or female) and both are equally likely each time, then the odds of boy coming out 18 times in a row is 1/2^18. I think that's 1/262,144.

Bob Ellison said...

Then again, it could be, perhaps, that the purple dye weighs less than the other dyes, or else somehow the purple ones are smaller. That could explain it.

Boy babies, I understand, are slightly more likely than girl babies. That seems sad.

CharlesVegas said...

Joe P, I get the probability of 19 boys born in a row as 1 in 524288 based on the binomial distribution probability mass function.

tim maguire said...

My parents had 9 kids. The first 5 grandchildren were all girls and my father began to despair of having a grandson.

The next 14 were all boys.

Pogo said...

"Put all the baby boys together from head to toe and they total about 30 feet. Their total weight is a combine 115 pounds of baby boy."

That's nothing.

In a 14 hour period in the US, there are about 1,938 abortions.

Put all the dead fetuses together from head to toe and they total about 401 feet. Their total weight is a combined 114 pounds of fetal tissue.*


*(data from Guttmacher)

CharlesVegas said...

tim, 5 girls then 14 boys has same odds as 19 boys born in a row.

Pogo said...

And the chances of that are 100%!!

Bob Ellison said...

In China a hundred years ago, the odds of a boy birth were probably pretty high.

LarsPorsena said...

19 boys and no girls, y?

James said...

More like one in 131,000.

You need the next 18 births to be the same gender as the arbitrarily-selected first birth, but since you don't care about whether the run of 19 is all boys or all girls, you add both probabilities together. 1/(2^17) = 1/131,000

Bob Ellison said...

I have four boys and no girls. But my dogs are all girls. Who will take care of me in the future? Maybe I should change my name to Julia. I don't think I'm willing to go farther than that.

MikeR said...

Barry Sanders is correct that this (starting with either a boy or a girl) has one chance in 262,144. But the next thing you have to do is see how many times the trial is repeated. In other words, in all the hospitals in the US, say for a year, how many sequences of 18 babies are there? I assume the story would have been published regardless of which baby was first. So if a particular hospital delivered two thousand babies that year, they have about 2000-17 chances to get an interesting string of 18 babies in a row. Then there are a lot of hospitals, and each of them could have been the lucky one. I don't have the numbers, but if I did I could work it out. I'm not sure this is so terribly unlikely when you have millions of babies born each year.

Also, I assume the newspaper wasn't that fussy. They probably would have published the article even if there had been eighteen alternating boys and girls, or maybe twenty-four. That would also be kind of neat, right? Or any number of other long interesting sequences. So part of understanding the probability of this happening is deciding what "this" means. There are a lot of unusual sequences that could have prompted such an article, and the combination of allowing any of them makes the total more likely yet.

That said, a long enough sequence of boys would be unlikely enough to be miraculous, or to prompt some biologist to try to discover a new fundamental fact about babies. I'm not sure this was long enough for that.

Bob Ellison said...

James, that sounds right. I hope you know what you're doing, since I don't.

So: newsworthy. Makes one wonder: how did this happen? Statistics say it should happen, but they don't say it's obvious. It's either interesting news because it's such an outlier, or else it's interesting news because something might be going on. Which is it?

Pogo said...

I'm just amazed they went to the trouble of laying all those baby boys from head to toe just to measure them.

First, they squirm and cry. Then, the tape measure is hard to hook on the foot of the last one, and keeps snapping back, hitting kids on its way back into the dispenser.

To get a sense of the number, they could have said And this would fill 25 suitcases, or 24 if you knelt on them.

A for effort though.

Dante said...

Your intuition that it is not news is completely wrong. As an example, there is roughly 1 in 500,000 chance of getting 19 boys. But, there are 19 times as many chances of having 18 boys and 1 girl. And there are 92,378 times as many chances of having 10 boys and 1 girl.

It's a part of math called combinatorics.

Now, if you were to say, I want 1 boy, 1 girl, 1 boy, . . . in that specific order, then yes, the chances of THAT are the same as all boys. Except that more boys than girls are born, but whose going to quibble when you are off by nearly five orders of magnitude base 10.

miss j said...

It is true that the specific sequence 19M is just as probable as the sequence MFMFMFMFMFMFMFMFMFM and just as probable as FMFMFMFMFMFMFMFMFMM. However, both of the latter (and 92376 other sequences) produce 10M9F.

However, you used the word "combination" and words matter. The combination of 19M only happens 1/524288 times (i.e. approximateily 0.0002%) while 10M9F happens approximately 17.6% of the time. Big difference. Indeed, it's quite interesting because unusual occurrences are interesting.

Of course, the combination of at least 19 male births out of N births becomes increasingly common as N increases. In fact, even the sequence of 19M becomes more likely if you consider N>19. Why, for N=20, there are twice as many ways to get 19M!! I believe this is what you might have been trying to say when you spoke of the "period you might define." I'm not quite clear of your meaning. "Period" is typically used as a measure of time and not to describe events that can be quantized (i.e. number of births). Naturally, the period over which this was measured is not the relevant piece of information. It doesn't matter if the 19 births took place over 3 days or 5, it still is an unusual combination.

In any case, if this is your argument, I believe this was quite widely discussed in the media several years ago because of Freakonomics. Unfortunately, I don't remember the reference because I remember the concept from elementary school when we all learned about Pascal's Triangle.

It's funny to me that people struggle with everyday concepts and definitions when they are such a part of our basic education.

Dante said...

of having 10 boys and 1 girl.
of having 10 boys and 9 girls.

rhhardin said...

They could be all from the same milkman. The man determines the sex.

Pogo said...

"The man determines the sex."

On what planet?

Oh, the chromosomes, not when.

Dante said...

That's MISS (not mrs) j, right =)

edutcher said...

elk had my thought.

What makes it interesting is the dirty little secret that abortion is used to gender select which babies are born.

I'm sure the feminazis swooned, also.

SeanF said...

Ah, yes, the joys of probabilities.

You randomly meet a person, and learn two things about them - they have two children, and at least one of their children is a boy.

What are the odds their other child is also a boy?

(Let's just assume an exact 50/50 gender birth rate, shall we? :) )

Dante said...

SeanF: 50%. If they are talking about their first born, their second born is either a girl or a boy. If they are talking about their second born, their first born is either a girl or a boy.

Now for you, surprisingly I've heard this was discovered last century. 3 doors, one has Miss J., the other a couple of guys. You pick door one, one of the two remaining doors is revealed to you, and it's a guy. Do you switch or not, and why? Yes, enjoy.

Sigivald said...

it's no more unusual than any other combination of boys and girls born anywhere else over any other period you might define.

Not so.

I mean, you're right that it's not really news, and that there's certainly some cherry-picking around the edges.

On the other hand, it's not "no more unusual than any other combination", any more than "getting Heads 19 times in a row with a coin toss is no more unusual than any other outcome".

(It's just as likely as any specific distribution of heads and tails in order, but not as likely as any other sum of outcomes.

There's a reason the normal distribution looks like a bell curve, not a flat line, after all.

As far as I know, the distribution of births by gender and time at any given American hospital will be more or less a normal distribution, so getting 19:0 is less likely by far than 10:9 or 11:8, for any given 19 births.

To repeat, as you said, the fact of 19 and the 14 hours suggests that immediately before and after there were girls born.

It's not amazing, just fairly uncommon. [Wolfram Alpha suggests odds of 1 in half a million for any random 19 births, but of course we can aren't looking at random groups of 19 births...])

MadisonMan said...

That's part of why I bought a few tickets in the current Powerball lottery.

It may interest you to know that the winning ticket was sold in Wisconsin today. On Regent Street, in Madison, actually. At about 9:30 AM.

The purchaser then bought a donut.

What are the odds of that happening!?

Lem said...

Did Nate Silver give odds on that probability?

Nate Silver went on all five Sunday Morning Talk Shows and said that the births were spawn in reaction to a porn video.

Drago said...

Sigivald: "There's a reason the normal distribution looks like a bell curve, not a flat line, after all."

Racism?

Lem said...

MM comes out of the Lottery Closet.

Just want you to know, I'm very proud of you.

PETER V. BELLA said...

It is a sexist hospital.

Eustace Chilke said...

Intuition about the liklihood of this happening tends to be wrong for the same reason as in similar sorts of odds making problems such as; how many people must be chosen at random to yeild 50/50 odds that two of them have the same birthday. The answer is 23. The number is as low as it is because no specific date is specified for the birthday match.

With the string of boy births the starting point is similarly open. The only condition is that 18 in a row be run up, of either sex it can be assumed, starting any time. Not particularly remarkable that it happened nor that someone would think it was remarkable.

Dante said...

What are the odds of that happening!?

The most odd thing in my life was when my kids were born. What are the chances of that? I mean, a big yellow bird, . ..

No but seriously, consider the survivors out of each generation are miniscule by themselves. How many generations back to the beginning?

The odds are so vanishingly small. Even if there were only a million generations, and your odds of surviving are only one in a million, that's one in a million to the millionth power. There aren't that many protons, neutrons, and electrons in the universe as that number.

Quaestor said...

Did Nate Silver give odds on that probability?

This is an example of a classic corundum of probability. Take anything than can exist randomly in any given number of states and strings of one state are to be expected.

For example, let's say I give you the task of experimentally flipping a fair coin one hundred times and recording your results. Now suppose you turn in your supposedly random data and it looks like this:

1) H
2) T
3) H
4) T

... and so on, heads and tails usually alternating with occasional strings of two or three heads or tails, with a final total of 50 heads and 50 tails. My reaction would be to suspect you've cheated and not done the experiment at all.

Actually, one hundred trials is far too small a sample to accurately reflect the 50/50 probability of heads or tails with a fair coin. In one hundred trials a slightly skewed results like 56/44 or 57/43, with long strings of the one outcome is more likely than 50/50.

I once opened a bag of Skittles and poured a sample into my hand. All five came out purple.

The odds of this happening are exactly the same as any other combination of Skittles, assuming that the average bag of candy contains roughly equal proportions of the available flavors. What would be remarkable would be getting the same flavor initial handful out of two successive bags of candy.

Then again, it could be, perhaps, that the purple dye weighs less than the other dyes, or else somehow the purple ones are smaller. That could explain it

I doubt it. Let's assume for arguments sake that different flavor Skittles have different masses, greens are heaviest, blues lighter, yellows lighter still, and purple lightest of all. Let us also that confined in a bag and given certain amount of agitation the candy would sort itself into layers, say X number of agitations would achieve maximum order in the bag. However, the next disturbance, agitation X+1, must promote greater randomness, X+2 more randomness, X+3 even more, and so on. The Skittles are going to oscillate between various degrees of randomness and very rare states of order.


JRPtwo said...

If I’m doing the math right, a run of a bit more than 19 all boys or all girls is expected at some hospital in the US each year. The expected longest run of heads or tails from flipping a coin is about log base 2 of the number of flips, plus or minus 3. See gato-docs.its.txstate.edu/mathworks/DistributionOfLongestRun.pdf
There are about 4 million births a year in the US, so if they were all at one hospital, the longest run would likely be about 22 plus or minus 3, since 2^22 is about 4 million. The top 1000 hospitals have about 70% of the births, so partitioning the string into 1000 big parts takes care of most births and seems unlikely to cut the longest string.

Quaestor said...

Oops, I meant to type conundrum. Betrayed by spellcheck once again.

Bob Ellison said...

Quaestor, I don't think you understand probability. The odds of any combination of Skittles are equal, if you suppose the Skittles have no color. The color makes the odds.

This is a popular problem with probability. People don't understand things like standard deviation and unlikeliness. It's unlikely that I will win this week's Big Red Ball lottery, whatever it's called, but if I win it nonetheless, that does not change the probability.

DCS said...

Considering that my local CBS affiliate produced the story, I'm not surprised at the complete idiocy of the report. WCCO also sent a reporter to cover the "strike" by union organizers at Walmart on Black Friday and failed to ask any of the picketers if they actually worked at Walmart. A rival station did, and established that it was not a work stoppage but a photo op for the union.

Bob Ellison said...

MadisonMan, I am sorry to have missed your earlier message. You are mistaken, sir. Check the numbers. Perhaps a few dollars were won. In fact, the lottery was won today by some guy at a Wawa. Ha ha! He intends to use part of his winnings to buy a furnace for his cat.

Bob Ellison said...

Re-reading the Professor's original post, I think she doesn't understand probability either.

Chip S. said...

I cannot believe that this post has been up for almost 4.5 hours and not one single Althouse geek has run a Monte Carlo simulation.

Bob Ellison said...

It's a sad day for Althouse geeks.

Chip S. said...

My excuse is that I'm still trying to figure out how the maximum likelihood estimator doesn't yield the maximum likelihood estimate of the mean of the binomial distribution.

Sam L. said...

When you choose the facts, and beat them severely with at least 70% accuracy, zey vill do vot you vant!

Saint Croix said...

They are probably doing a lot of doggy style and/or cowgirl in Minnesota.

Swim, sperm! Hurry!

Cold weather = deep penetration = boy.

I don't know if it's true, but it's on the internet.

MisterBuddwing said...

What? Nobody remembers that SNL sketch in which Weekend Update interviews an obstetrician who's been delivering nothing but girls for the past 20 years? (It turns out he was surgically "correcting" the boys.)

SeanF said...

Dante, the odds that the other child is a boy are 1/3, not 1/2. If you know the first child is boy, it's 1/2; if you know the second child is a boy, it's 1/2. But if all you know is that there is a boy, it's 1/3.

Your question about the doors is vaguely worded (as it usually is when people ask it). If the opening of the unpicked door was a random act, then it makes no difference whether you switch or not. If a door with a guy behind it was chosen intentionally, then it's to your advantage to switch. Either way, though, switching won't hurt your odds.

Saint Croix said...

I hate math, but I love sex.

Saint Croix said...

Ya'll turned a sex thread into a math thread. Just sayin'.

gsgodfrey said...

There seems to be a misunderstanding about combinations versus sequences. For 19 consecutive boys, there is no difference because there is only one sequence that produces a combination of 19 boys and 0 girls. For 10 boys and 9 girls, there are many sequences that produce that particular combination.

Same for the Skittles. Grabbing the first five from the bag (and assuming there are five possible colors), there are 5!=120 permutations in which each Skittle is a different color, but only one that is all purple (or five that have the same color). In other words, it's 120/5=24 times more likely for a handful of five Skittles to each be a different color than the same color.

Assuming the probability of a boy birth and a girl birth (or Skittle color) is the same, all sequences are equally probable. However, all combinations are not.

Give me a few more minutes and I think I can produce the expected number of births required to get a run of 19 consecutive boys (without resorting to Monte Carlo simulation).

Lem said...

In everything, it is the act of paying attention, of measuring, that is unusual.

Take the lottery for example..

It may seem remotely unlikely for one person to win more than once and yet that is exactly what has happened.

gsgodfrey said...

The expected number of births required to get n consecutive boys can be computed from a simple recursion.

Let E{n-1} be the expected number of births to get n-1 consecutive boys (that is, the last n-1 births are all boys and that is the first time that sequence appears).

We can compute E{n} given E{n-1} as follows. Consider the next birth (the E{n-1}+1st birth). If it is a boy (with probability p), then we have success and can stop. E{n-1}+1 births were required to produce the first sequence of n consecutive boys.

If the E{n-1}+1st birth is a girl (with probability 1-p), then we need to start over, meaning that E{n-1}+1+E{n} births will be required to get n consecutive boys.

The recursion, then, is

E{n} = p*(E{n-1}+1) + (1-p)*(E{n-1}+1+E{n})

Combine the E{n-1}+1 terms together

E{n} = E{n-1} + 1 + (1-p)*E{n}

p*E{n} = E{n-1} + 1

E{n} = E{n-1}/p + 1/p

For p=0.5, we have E{n} = 2*E{n-1} + 2

Starting with E{0}=0, we get E{1}=2, E{2}=6, E{3}=14, ...

In general, for p=0.5, we could show that

E{n} = 2^(n+1) - 2

The expected number of births required to have 19 consecutive boys is E{19} = 2^20 - 2 = 1,048,574.

For any particular hospital, that is rare (and newsworthy). However, given 4M births per year in the U.S., it is not that unusual to have happen somewhere in a given year.

madAsHell said...

It's a war on womyn, I tell ya'.!!!

Who's going to pay for all the contraceptive??

Saint Croix said...

The recursion, then, is

E{n} = p*(E{n-1}+1) + (1-p)*(E{n-1}+1+E{n})

Combine the E{n-1}+1 terms together

E{n} = E{n-1} + 1 + (1-p)*E{n}

p*E{n} = E{n-1} + 1

E{n} = E{n-1}/p + 1/p

For p=0.5, we have E{n} = 2*E{n-1} + 2

Starting with E{0}=0, we get E{1}=2, E{2}=6, E{3}=14, ...

In general, for p=0.5, we could show that

E{n} = 2^(n+1) - 2


You're wrong! They're doing the cowgirl in Minnesota, damn it.

Dante said...

Seanf: I gave you my reasoning:

1/2 the child they are talking about is the first child, the other child is either a girl or a boy, 50/50.

1/4 B B
1/4 B G

1/2 the child they are talking about is the second child, the other child is either a girl or a boy. So you get:

1/4 B B
1/4 G B

So you get 1/4 B B + 1/4 B B = 1/2 B B

and 1/4 B G + 1/4 G B = 1/2 1 girl + 1 boy unordered.

Please explain how your math works out.

Saint Croix said...

Please explain how your math works out.

It's geometry. Or physics. Something. She's on top. Deeper thrusting. It's really not that complicated.

Saint Croix said...

1 + 1 = 1

and then, 9 months later

1 + 1 = 3

check my math, I think that's right.

Saint Croix said...

Unless we start counting the unborn. Then it's 1 + 1 = 2.196 You got to break out the decimals. This is what got the Supreme Court in trouble.

gsgodfrey said...

Dante: You are double-counting. The parent's original statement is not whether their first child is a boy or their second child is a boy. The statement is simply that at least one is a boy.

There are three equally likely possibilities: BG, GB, BB, only one of which has the other child a boy. Answer is 1/3.

Suppose the followup question to "at least one is a boy" is "which child are you thinking of" (which is what your logic is based on). The BG parent has to say first (100%), the GB parent has to say second (100%), and the BB parent has a 50% chance of saying first and a 50% chance of saying second. Thus, if the answer is "first", then there is a 1/3 chance that the second child is a boy (BG must say first all of the time, and BB only half of the time, so it's a 2G:1B ratio).

Your answer is 1/2 because you assume wrongly that BG and BB are equally likely to answer "first". That's the double-counting problem. There is a big difference between asking "Which child are you thinking of" and "Is your first child a boy".

Tyrone Slothrop said...

A lot of people are throwing out a lot of wrong probabilities. Odds of 19 boys in row are 1 in 2^19, or 1 in 524288. Long odds, but not astronomical.

Tyrone Slothrop said...

Odds of two boys in a row--

gg
gb
bg
bb

1 in 4

odds of three boys in a row

gbb
gbg
ggb
ggg
bgg
bgb
bbg
bbb

1 in 2^3 or 1 in 8

Odds of 4 boys in a row--

gggg gggb ggbg ggbb
gbgg gbgb gbbg gbbb
bbbb bbbg bbgb bbgg
bgbb bgbg bggb bggg

Odds are 1 in 2^4 or 1 in 16-- and so forth.

Odds of a string of any length of boys (or girls) is 1 in 2 raised to the power of the string length.



Chip S. said...

Tyrone, your answer presumes that the story wouldn't be equally newsworthy if it involved 19 girls. If you don't care whether it's a run of boys or girls, then you're only looking for a run of 18, so Barry Sanders20 got it right.

But the question of how unusual this is really applies at the level of the TV station's reporting area.

With 4 million births/year, the expected number of occurrences of streaks of 19 births of either sex nationally is about 15 [= 4M/2^18].

What are the odds of such a streak happening in Mpls/StP?

According to the Statistical Abstract, MN accounts for about 1.7% of total US births. Le't suppose that about 2/3 of those occur close enough to this TV station's range to be newsworthy to it, so it would've reported on any unusual birth streak occurring among approximately 40,000 annual births.

In that case, this streak has an annual probability of about 40,000/2^18, so it could be expected to occur on average once every 6.5 years or so (avg. time per event = inverse of prob. per unit of time).

That seems newsworthy, but not amazing.

Chip Ahoy said...

And then even with those astounding odds in each case were you to ask "what are the odds of the next baby being a boy?" All of your calculations above are for naught because you must say 50%. And the next baby and the next baby after that and so on to the 14th baby and even the 15th baby.

So I'm told.

I kept telling people if you have 10 flips of a coin in a row all the same result the chances of the next flip to differ are vastly increased because of the preceding string is even more unlikely to continue and the answer is always, "No, Chip, your problem is you never took statistics class or you wouldn't say something that stupid."

Chip S. said...

If I see a coin come up heads 10 times in a row, I figure that there's a more than 50% chance of heads on the next toss, b/c I start to suspect that it's a two-headed coin.

Consider the roulette wheel:

In the early 1990s, Gonzalo Garcia-Pelayo believed that casino roulette wheels were not perfectly random, and that by recording the results and analysing them with a computer, he could gain an edge on the house by predicting that certain numbers were more likely to occur next than the 1-in-36 odds offered by the house suggested. This he did at the Casino de Madrid in Madrid, Spain, winning 600,000 euros in a single day, and one million euros in total. Legal action against him by the casino was unsuccessful, it being ruled that the casino should fix its wheel.[9][10]

To prevent exploits like these, the casinos monitor the performance of their wheels, and rebalance and realign them regularly to try to keep the result of the spins as uniform as possible.

Carl said...

I don't think you can legitimately say breaking up the 4 million births in the US into separate hospitals makes no real difference. If a sequence of 19 boys occurred with 4 at one hospital, 5 at another, 2 at still another, and so forth (adding up to 19), this would not be newsworthy. The key aspect of this story is that it happened at one hospital.

The rate of birth at this hospital is apparently 19 births in 62 hours, or about 2685 births per year, assuming this represents a good average rate and births take no holidays. So the question becomes: what is the probability that for THIS hospital, 19 (or more) boys in a row are observed one year?

The probability that NO run of k boys occurs in n births is F(k)(n+2)/2^n, where F(k)(n) is the nth k-step Fibonacci number. Writing a little careful code to calculate F(19)(2685+2)/2^2685 gives 0.997459, or equivalently a probability of 0.002541 (or a bit better than 1 in 400) that at least one sequence of 19 or more boys happens for this hospital in a given year.

That's a long way from the reporter's casual 1 in 500,000 odds, but it's still pretty rare, something that isn't likely to have happened within the memory of any employee. So I think it's "newsworthy" for at least the Odd Things page.

whoresoftheinternet said...

Thank goodness.

What we don't need in this world is any more feminazi broads who kill their own children and sacrifice everything...even the truth...to Mein Obama.

Michael said...

I would say that the binomial statistics here are all not asking quite the right question: The question should be what is the expected longest run in N births?

As it turns out, given a random set of 524,288 births with equally likely boys and girls, you would expect to have a run of 19 births of either boys or girls (with a standard deviation of about 1.44 births). CharlesVegas got the right number for the wrong phenomenon.

Depending on which study and country you consult, the natural secondary sex ratio for humans is around 104:100 or 105:100, meaning you'd need about 661,000 births to get a run of 19 boys -- but 1.7 million births to get a run of 19 girls!

There are only about 4 million births per year in the US, so it's not something that will happen very frequently -- and it would happen most in busier hospitals, where the event might pass unremarked.

Paco Wové said...

"if you have 10 flips of a coin in a row all the same result the chances of the next flip to differ are vastly increased"

Chip Ahoy, the anti-Bayesian statistician.

Ann Althouse said...

If you're quarreling with my statement, you must pay attention to my statement: "it's no more unusual than any other combination of boys and girls born anywhere else over any other period you might define."

Every combination of males and females is equally likely.

(The only real quarrel I see is that in reality very slightly more boys are born than girls.)

SeanF said...

Dante, Gsgodfrey explained the error in your math. You're right that half of them would be thinking of their first child, but boy-girl parents are more likely to be thinking of their first child (100%) than are boy-boy parents (50%). Remember that two-thirds of the possible families have a first-child boy, and two-thirds have a second-child boy - it's not 50-50 in this case, because we've eliminated all the girl-girl families, and the boy-boy families are in both groups.

But now consider this - if you know at least one of their kids is a boy because he's standing right next to them, then the odds move to 50%. Half the boy-girl parents (and half the girl-boy parents) would have a girl next to them, so you can eliminate them from your possibilities.

Probability is dependent on what you know. Same with your question about the doors.

SeanF said...

Ann Althouse: Every combination of males and females is equally likely.

Every permutation is equally likely, not every combination. If you flip a coin four times, you are more likely to get two heads and two tails than to get four heads.

HHHH is just as likely as HHTT, and just as likely as TTHH, and just as likely as THTH, etc. There are six permutations which result in a combination of four heads and four tails, but only one permutation which results in a combination of four heads.

The quibbles are about the distinction between permutations and combinations.

Ann Althouse said...

"That's a long way from the reporter's casual 1 in 500,000 odds, but it's still pretty rare, something that isn't likely to have happened within the memory of any employee. So I think it's "newsworthy" for at least the Odd Things page."

But 500,000 birth sequences are occurring all the time, so the 19-in-a-row boys one is utterly ordinary. It only seems extraordinary because the other occurrences don't impress you.

MadisonMan said...

The news item

There were 17 children born at the Hospital today in this sequence: Boy, Boy, Girl, Boy, Girl, Boy, Boy, Girl, Girl, Girl, Girl, Boy, Boy, Girl, Boy, Boy, Girl. Hospital workers say they cannot recall such a precise ordering of boy/girl births in the past many years

would be awesome to read someday, if only for hilarity value.

Saint Croix said...

I can't believe I am the only one who thinks sex is more important than math. You're all off topic!

Clyde said...

If it were Beijing or Mumbai, it would be less surprising. There are a lot of countries with very skewed sex ratios at birth due to sex-selective abortions, even if such are frowned on by the authorities. This is especially true in China, with its "one-child" policy and no retirement safety net for old people other than support from their children. Sons are more highly prized in such a society than daughters.

Carl said...

It only seems extraordinary because the other occurrences don't impress you.

Althouse, I know this observation seems keen to you. Like a bar bet an AP Stats student wins by pointing out the odds of drawing 8H 6C 2D 5D QC (or some other worthless hand) in poker are exactly the same as drawing AH KH QH JH 10H.

But for those who've penetrated a bit further into maths, the epiphany that all sequences are equally unique occured in Chapter 1, quite some time ago. At least in my case, that's why it escapes comment whilst pondering the interesting question of what, exactly, are the relevant odds that underly the story -- whether, considered as the author obviously meant it, this is or is not a genuinely rare thing. It is, although perhaps not quite as rare as the reporter thought, once he turned to the numbers.

In fact, one of the more interesting aspects of this is that my impression is that both the reporter and the participants seem to have a better intuitive grasp of the odds than mathematical. They are not reporting the event with the kind of genuine amazement with which they would greet something that has an actual 1 in 500,000 chance of happening (e.g. the natural birth of quadruplets, or one woman having twins twice). It's not front page news, just "how interesting" back page filler.

JRPtwo said...
This comment has been removed by the author.
Dante said...

SeanF: I looked this up, as actually at one time I spent a day deriving BOTH ways of looking at the problem. It's tricky, no doubt.
In one you have a random family:

GG
BG
GB
BB

and saying "I have at least one boy" elimates GG and so you have the BG GB BB possibilities, or as you say, 1/3 of two boys.

However, if you are a father, and you fathered two children, and only one of the women tells you "I had a son", and you do not know the sex of the other child, then the chances of the other being a boy is 1/2. In this case, they are completely random events. So in this case, a person saying "I have at least one son" gives you fifty-fifty chances. Pretty odd, eh?

JRPtwo said...

Ann, even if we agree that this particular sequence of 19 is just as likely as any other (if it’s random) and recognize it was cherry picked to start and end on a run of boys, you still need more to conclude it’s not “amazing.”

Your reasons would apply equally to a sequence of 500 boys, and that would be amazing (or surely not random).

Dante said...

Every combination of males and females is equally likely.

It depends on what you definition of "combination" is. Here is a webster definition: any subset of a set considered without regard to order within the subset

Using that definition, you are incorrect. It would have been much better to say "Any sequence," but you didn't. And you are a lawyer. You are either guilty of not expressing your thoughts well, devaluing your lawerly ability to use the english language, or you are simply wrong.

Given that you have admitted to lying in order to not appear ignorant, I think the later has higher probability.

Ann, admit you made an error. Really, it's important for you to understand you too are just a human, after all.

Dante said...

It only seems extraordinary because the other occurrences don't impress you.

That's actually also not correct. There are ways of determining within a sequence whether or not the sequence is so odd as to indicate favor for one or the other, thereby indicating the underlying probabilities are not what we think they are, with given probability.

So your supposition that it's the same as any other sequence is not, in fact, correct either. As this sequence could indicate somethings up.

They do this in climate science all the time. They say "Hurricane Sandy is improbable, therefore the underlying probabilities are changing." Now, having looked at the data, I happen to disagree with this, but it is a valid mathematical technique. In fact, I think it was used by some woman to win the lottery a bunch of times.

SeanF said...

Dante: However, if you are a father, and you fathered two children, and only one of the women tells you "I had a son", and you do not know the sex of the other child, then the chances of the other being a boy is 1/2.

Yes, but there you know that a specific child is a boy. It's the equivalent, in my original scenario, of knowing that the person's first child was a boy.

If someone else who knows both women tells you that at least one of them had a boy (but doesn't specify which one), the odds of the other also having had a boy are 1/3.

Really, the question is, "If they're not both girls, what are the odds they are both boys?" And that's different than, "If this one's a boy, what are the odds that one's a boy?"

Dante said...

Sean, actually your problem isn't well stated, and one correct answer is indeed 50%. Consider your question assumes the person knows the sex of both children. If he does not, you have the ordering to eliminate one combination.

All you can say in answer to your question is it's either 50% or 33%.

Anyway, point taken. Puzzles are fun.

SeanF said...

Dante, you're the one assuming.

Probability is about knowledge. If you know that a card was randomly selected from a regular deck, there's a 25% chance it's a spade. If you also know it's black, the odds are 50%. Nothing changed but your knowledge.

In my question, I told you exactly what you know - births are split 50/50 by gender, the person has two kids, at least one of them is a boy. Knowing only that, the answer is 1/3.

In order to get an answer of 50%, you must assume additional knowledge - which kid is a boy (note that even in your example, it doesn't matter if the person to whom you're speaking only knows the gender of one kid; it only matters if you know that).

But if you're going to assume additional knowledge, you might just as well assume you know the other kid is also a boy and say the answer is 100%.

The way I stated the question, the only answer is 1/3. To get other answers, you have to make unwarranted assumptions.