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The are 45 people in my extended family. Only my sister was born in October. What are the odds of that occuring?

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You're basically looking for what are the odds that all 45 people do not have October birthdays, so this should be fairly straightforward. Depending on how precise you want to get it. If we just ignore that months have different lengths, it would be:

$$\bigg({11 \over 12}\bigg)^{45} = 0.0199 \qquad{\rm or}\ 1.99\%$$

If we look at October having 31 days vs the other 334 days of the year (ignoring leap years) it would be:

$$\bigg({334 \over 365}\bigg)^{45} = 0.0184 \qquad{\rm or}\ 1.84\%$$

All of the above assumes that people are born in equal distribution throughout the year, but in reality they aren't. You could look at some census data to see which months have more births and which ones have less then come up with a way to weight it accordingly. I have a feeling you aren't looking for this layer of precision and are merely curious at an off-the-cuff answer.

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  • $\begingroup$ I think, to make it even more exact, one also would need to incorporate the correlation. Say if someone was born on 1st of November, there is a chance, that he has a sibling born on the same day (e.g. twins). $\endgroup$ – jottbe Oct 14 '19 at 13:23
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    $\begingroup$ Please state all the assumptions you have made in order to carry out these calculations. One that is extraordinarily important is independence of birth months. Then, please revisit your calculations, because they are inappropriate for the question: you should be computing the chance that one or fewer people were born in October, not that zero people were born in October. You will get a much larger answer (around 9.5%). $\endgroup$ – whuber Oct 14 '19 at 13:47

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