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I bought one dozen fertililized eggs, incubated them and hatched 11 roosters and 1 hen! This was just the opposite of what I'd hoped. What were the odds of this happening?

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Congratulations, but it depends on many factors. I don't think that the probability of male or female hatchlings is a half. It could well depend on temperature besides many other factors. –  BlackAdder Feb 21 at 1:07
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4 Answers

Depends on how hot the incubator was:

There was no difference in sex ratio between chickens hatched from eggs stored at 60 and 80°F, neither departing significantly from 50 per cent. pullets. However, storage at 40°F produced significantly more pullets than cockerels, 54.6 per cent. of the chickens being pullets

Taken from this study: http://www.publish.csiro.au/paper/AR9600664.htm

So for 50% it's $${12 \choose {1}}*2*(1/2)^{12} = 3/512 \approx 0.006%$$

For 54% pullets or female chickens,

$${12 \choose {1}}*2*(.546)^{11} *(.454) \approx 0.01%$$

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If you think the innate odds are $50\%$ for each, the chance of getting a split of $11-1$ (either way) is $2{12 \choose 1}(\frac 12)^{12}=\frac {24}{4096}\approx 0.006$ You were either unlucky or the odds are not $50-50$

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Assuming that the probability of getting a hen and a rooster is equal, it's follows a binomial distribution with $P=0.5$ of success:

$${12 \choose 1}\cdot(0.5)^{12-1}\cdot(0.5)^1=12\cdot(0.5)^{12}\approx 0.3\% $$

This is the chance of getting 11 roosters and a hen.

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The likely reason you got 11 roosters to 1 hen is that you chose 11 eggs containing roosters. Their gender is estalished before laying. The big breeders want hens to keep up their egg production and because female meat chickens fetch a higher price. So they keep the female eggs -those they think are female - and sell the suspected males. I found a website suggesting definite male eggs have a pointier end, whereas definite females do not, and are more perfectly oval. The ones in between are apparently harder to determine. I'm guessing you bought your dozen fertilised eggs from a breeder who knew how to retain as many female eggs as possible - or sell them to more knowledgable buyers. I'm sure there is a mathematical ratio once you know how many eggs, on average, contain hens or roosters. But as I used to say to my (long-suffering) maths teachers: "How is this ever going to help me in life?!" Finally, on this occasion, I think lateral thinking wins the day. Not that I would encourage you to buy more eggs. Your experience serves to remind me of how many potentially characterful roosters are killed just to allow a human to indulge a culinary habit or preference. You can eat very well and be healthier without other species flesh, eggs or breast milk in your diet. However, it may be morally justifiable to buy/rescue 18mo layer hens when they are being sent to slaughter. Hens lay for around 5 years, slowing from around 18months, and can live 10-15 happy years when well-cared for.

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