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This question already has an answer here:

(retrying my phrasing)

What are the odds of having 4 (and only 4) naturally conceived, naturally birthed, singleton children (with the same and only mate) on a given day of the week (ie, Wednesday)?

Equations with explanations appreciated. Thanks

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marked as duplicate by gung Aug 11 '18 at 16:22

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    $\begingroup$ Another question you might like to consider is "what is the probability of four out of four kids all being born on the same day of the week?". This might be more relevant if you would consider it equally amazing if the kids were all born on a Thursday, or all born on a Friday. $\endgroup$ – markseeto Apr 16 '16 at 23:20
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    $\begingroup$ Doctors do C-section Wednesday mornings, hit the links in the afternoon (real story is that birth day of week is not equal across days of week, and may not be independent for same doctor or hospital across multiple kids). Also, do you really want to know probability of all kids being born on Wednesday, or do you want to know probability that all kids are born on the same day of the week? Also, if you go "hunting" after the fact for unlikely events or records, it's easy to find them - that's a form of massive simultaneity bias. $\endgroup$ – Mark L. Stone Apr 16 '16 at 23:21
  • $\begingroup$ If day of week that kids are born on is really independent (across your kids) and identically distributed with probability 1/7 (none of which are likely true), then the probability that all 4 kids are born on Wed. is $(1/7)^4$, which can be viewed as the probability of 4 successes in a Binomial Distribution with 4 trials, each of which has probability 1/7 of success, defined as "kid born on Wednesday". $\endgroup$ – Mark L. Stone Apr 16 '16 at 23:31
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    $\begingroup$ One important point, is that you have picked out a pattern as a coincidence, and then asked for the odds of that particular pattern, but you should really ask what are the odds of any identifiable pattern, e.g., all born on the same day, or on consecutive days, or on days beginning with a T or an S, or in the same month, or on the same day of the month, or in consecutive months, or or or. The odds of some kind of coincidence are actually quite high. $\endgroup$ – Kelvin Apr 17 '16 at 1:13
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    $\begingroup$ @Kelvin , your most recent comment is what I was trying to get at in the last sentence of my 1st comment. As for quadruplets, that blows the independence assumption across kids out of the water. Of course, if you want to game the system, you can fly in a plane near the international dateline, and no matter what time of day, by going to one side or the other, you have a choice of 2 days of the week. $\endgroup$ – Mark L. Stone Apr 17 '16 at 1:18

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