# How much evidence to suspect the "dice are loaded"?

We all know the odds of giving birth to a boy are roughly 50%. But that number is for the population in general and doesn’t necessarily apply to a particular couple.

First question, do some men make significantly more girl sperm than boy sperm or vice versa?

Second question, assuming the first is true, after how many births of one particular sex would you have statistically significant evidence that the man’s “dice are loaded”?

Example:

A man fathers one boy, totally normal. He fathers two boys, still totally normal. What if he fathers 20 boys and no girls? Is it now more likely a run of boys from a man producing normal sperm, or is it more likely that he’s not producing girl sperm? If not 20, what about 100, how many does it take?

I’m not really interested in the biology of this but feel free to address that if you wish. I’m more interested in the statistics, change the scenario to coin tosses if that makes the statistical angle clearer.

Well FYI, just in terms of the numbers, if the odds are 50/50, then the chance of 10 out of 10 is about one in a thousand…the chance of 20 out of 20 is about one in a million.

To get a good answer you need good a priori probability estimates. If I took a coin out of my pocket and got 15 consecutive heads I’d think it just a coincidence. If the coin belonged to a stranger who was tossing with me for money, I’d get suspicious long before the 15th toss.

For the X/Y sperm question, you’d want to check whether such sperm variation had been researched. (I’ve read that the X/Y odds can be influenced just by timing.)

There might be other factors than sperm production.

I’m more suspicious than septimus: If I flipped a coin and for 15 heads in a row, I’d be very, very firmly convinced it was a two-headed coin.

Yeah, I know, incredibly unlikely events do occur. At the same time, statistics allows me to declare a highly unlikely event to be a violation of the null hypothesis, and, in my mind, 15 heads in a row pushes the H-null farther than I like.

So, how many heads in a row would it take to determine the coin is unfair? Or, at least reject the hypothesis that it is fair?

I admit it: I’m a relatively gullible guy!

But it’s good to keep Littlewood’s Law in mind. And 15 consecutive same tosses is only about 16,000-to-1 against … exactly the same BTW as a strict alternation: H T H T H T H T H T H T H T H

This is a falacy, not sure of its name but it’s sent people to prison, in real life completely independent uniformly distributed random variables are rare. And if they are not completely independent uniformly distributed random variables the maths above is incorrect.

The fact is in most situations once some rare event has happened to someone (of probability of N% in the general population) the chances are of that thing happening a second time is NOT N%. Most real life events (the example in OP being one of them) are not perfectly uniformly distributed, there is some causal factor that means if has happened once it is more likely to happen you once, it is more likely to happen again.

I guess to follow on my point above, and actually answer the OP.

The dice are ALWAYS loaded. Unless you are in a situation where someone has put a lot of time and money into creating a set of uniformly distributed, independent, random events (e.g. a casino), it is safe to assume the events you are looking are neither independent or uniformly distributed.

If someone has two boys, they chances they have third boy is significantly more than 50%. In most real life events doing simple probability like “probability one event is X so chance of N events is X to the power of N” is wrong.

That depends on largely on what probability of “coincidence” you’re going to accept. When people do statistical analysis they have to make that decision, and common choices are 5% and 1%. In other words, if you’re using a 5% level, and the probability of the event you observed happening by chance is 3%, you would conclude that something is unusual.

A couple of more points:

As GoodOmens pointed out, you need to decide what level of “weird” is going to trip your trigger. In science, we often use a 5% cutoff, but that’s completely arbitrary.

There’s also the multiple testing problem. The probability of one specific person having a string of sons is different than the probability of someone in a large population having a string of sons.

You cannot discuss this statistically without discussing biology, because there is not a single variable involved. If you talk about coin tosses, you can control and eliminate other variables, but with babies, you can’t. A run of 20 boys can mean a man is not producing girl sperm. It can also mean his wife can’t conceive a girl. It can mean he produces girl sperm, but for some reason, his are defective. You can probably assume something irregular is going on (but it might just be the law of very large numbers); however, that the man is not producing girl sperm at all is probably not your first go-to.

First, not wholly true. Odds of conceiving a boy are about 60 in 100, odds of giving birth to a full-term boy are about 54 in 100. Odds of couple with a 2-year-old having a boy are about 51 in 100. Odds of a couple with a 6-year-old having a boy are about 48 in 100.

It’s called “Greater vulnerability in males.” It takes more energy to make a boy, because girls are the basic model, and boys are the variation. Girls are a little stronger at birth than boys, and the difference doesn’t completely disappear for several years.

No. All men make significantly more boy sperm, but for some men the ratio may be abnormal, to the point that the effect is the same as producing more girl sperm. Or, a man may produce even more boy sperm than is typical.

Girl sperm are bigger and last longer in the woman’s body, while the boy sperm vastly out-number the girl sperm, and swim a little faster. If a couple has sex right when the woman is ovulating, the boys get to the ovum before most of the girls, and “win” by sheer numbers most of the time. However, if a couple has sex a day or two before the woman ovulates, and not again until after ovulation, by the time the ovum shows up, most of the boys are dead, but a lot of girls are still hanging around.

Also, IIRC (and I may be misremembering, so someone please correct if so), the shorter the time between ejaculations, the more the boy/girl gap closes, so if a couple is having sex everyday, and the man ejaculates more than once, a girl is more likely, while if the couple has just one incidence of ejaculation, timed right with ovulation, a boy is more likely. But it never gets to the point that girl sperm actually outnumber boys, because there are so vastly many more boys to begin with.

There are other variables. Some women have a condition where their uterus is more hospitable to one type of sperm or the other, so a father may have four boys and no girls because of some condition the mother has. This is unusual, though.

Having unusual proportions of sperm usually run in families, so if a man has three sons and no daughters because “his dice are loaded,” the chances are he probably won’t have a sister or a niece either. Heritability of over-producing girl sperm is more complicated, because it men who have only daughters don’t produce a son to exhibit the trait, so it’s going to be something recessive or X-linked, and teasing it out is much more difficult. It could also be a spontaneous mutation, or it could be due to something not genetic.

My point there is that you don’t just look at a man’s offspring when you suspect something. A man with two sons and no daughters is not unusual, and has every reason to expect a 50% chance of a daughter if he has a third child. OTOH, a man with two sons, three brothers, eight nephews, and no sisters or nieces can figure on having another son, and may want to look into reproductive technology or adoption if he really, really wants a daughter.

Since OP’s question has been answered (see first sentence of #3 ), I guess we’re good to go with jocular answers.

Wow! Talking of probability and all, just last night – out of the blue in fact – I had the urge to find that Cohen song on YouTube! I didn’t see this thread until this morning. I have to wonder if there’s something (maybe Gravitational Waves?) messing with Reality these days…

And the way things have been going, I’m sorry I didn’t get in on the Death Pool this go-round. (Of course I never have.)

As to the OP’s concerns I remember a Time/Life book on Mathematics from 30 or more years ago that had pictures of two families with at least a dozen offspring each, with one all girls and the other all boys. I forget the point the pictures were there to support but it was impressive none the less.

It reminds me of one of those “pointless proverbs”: Fool me once, shame on you; fool me twice, shame on you; fool me three times…

[off-topic] I’d recommend this live rendition higher. (I linked to the other just because it starts with the punch-line “Everybody knows the dice are loaded.”)

I was singing melancholy Leonard Cohen songs to myself 45 years ago. :eek:

I need to confess that I had seen this thread’s title before I opened it to read it, and that may have been what (instead of “out of the blue”) prompted me to find the Cohen video.

Until I checked just now, I thought I remembered the song being on the Natural Born Killers (1994) soundtrack, but I can’t find evidence of that. Maybe some other movie?

Why wouldn’t you have checked the coin early on to make sure it wasn’t two-headed?

Can you show me a cite for this? All I could find is that the odds slightly change. In the US, the one study I could find shows the odds of having another boy after having two is 52.3%, which is slightly higher than the 51% odds normally expected. (Although I can’t find the actual study itself, just that summary of it.)

FWIW, when composing that first post I thought of mentioning that I’d check the coin, but felt that admission would make my post too verbose. :smack:

In any event, those in the thread looking for a single number ignore the very concept of a priori probabilities. In deciding how many heads you need to see before suspecting the coin is probably biased it makes a big difference whether this is a coin you just got at the bank, or one produced at a magician’s convention!

To answer the question about the man who fathered 20 sons, one would first want to review studies on such gender biases.

Always take Mr. Data along with you: https://www.youtube.com/watch?v=zbkizy-Y3qw

RivkahChaya, thanks for the information on the biology. I was aware that boys were favoured initially and then had a higher mortality rate but didn’t know the details.

I guess another way of phrasing this question is this:

You have 3 billion coins, they all flip fairly and taken as a whole you will get a 50/50 chance of heads vs tails, however some unknown but small number have either two heads or two tails. You take a coin at random and start flipping it. How many coin tosses would it take to be reasonably certain that you have a two headed or two tailed coin?

It seems to me that to answer the question accurately, you need to know the number of double sided coins, then you can just compare the probabilities of picking one of those coins at random against having a run of heads or tails from a normal coin. So as stated above, with the number of loaded coins being unknown, is it possible to come up with any meaningful answer at all?