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

[TokyoBayer]

You need to define what reasonably certain is.

Getting heads 26 times in a row would be in the same order of magnitude as selecting one special coin out of a billion.

Getting heads 200 times in a row has a one in e[sup]60[/sup] chance e.g. 1 in 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 which we can assume the odds would be astronomically against that.

The fortunate thing about flipping coins is that it’s such as easy task then one can quickly take the test to where the odds are so miniscule that it would be ten times, one hundreds times, or a thousand times more likely that you’ve found the double sided coin.

So pick a number you want to live with and flip it that many times.

[septimus]

@TokyoBayer

Getting heads 26 times in a row would be in the same order of magnitude as selecting one special coin out of a billion.

This is a good way to think about the problem but your arithmetic is off.

Assuming a run of tails is considered just as surprising as a run of heads, the “26 same-face in a row” has odds of 2^25 -to- 1 against. That’s one in 33 million, not one in a billion.

Of course even biased coins will come up tails sometimes (unless they only have two heads); this would complicate calculations.

[Enginerd]

@Zeldar

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?

Pump Up the Volume

[TokyoBayer]

@septimus

Of course even biased coins will come up tails sometimes (unless they only have two heads); this would complicate calculations.

I was looking at Richard’s post immediately above mine. I’m not sure what you mean by even biased coins come up tails sometimes. He stated:

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.

So there are no biased coins.

[Richard_Pearse]

@TokyoBayer

I was looking at Richard’s post immediately above mine. I’m not sure what you mean by even biased coins come up tails sometimes. He stated:
So there are no biased coins.

Yeah, there are no biased coins, just a certain number that will only ever give one answer.

This question was sparked by the fact I have three girls and I was wondering how many girls I would have to have before you could conclude that I probably will only ever father girls. Added to this, there is a bit of a meme going around the piloting community that we tend to have girls because of radiation exposure at altitude or something. I haven’t looked into it at all but suspect it’s probably just a bit of confirmation bias happening (I know plenty of pilots with boys.)

Although the biology of it is interesting in itself, I became more interested in a clear cut statistical scenario like the coins I described above.

[TokyoBayer]

Here’s a list of men who have fathered more than 30 children. Were any of them to have all sons or daughters then you could safely assume loaded dice.

The math is fairly straightforward, (despite my making a dumb error) so it all basically comes down to where you want to draw the line.