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Combined, what was the weight of the babies at birth?

Adding the pounds we have 3+2+2+2+3+2+2 = 2(3)+5(2) = 16 lb. Adding the ounces we have 4+11+10+5+3+14+15 = 62. So the answer is 16 lb., 62 oz. But, 62 oz. = 4 lb., 14 oz., so 20 lb., 14 oz or 20.875 lb. is the total weight. The average weight is 20.875/7 = 2.98 lb. or 2 lb. 15.7 oz. or 1.35 kg. The median weight is Nate's, 2 lb., 14 oz.

If a woman gives birth to septuplets, what is the probability that four of them will be boys and three of them will be girls?

This is (essentially) the same as asking for the probability of four heads in seven tosses of a fair coin. (Girls are slightly more likely to be born, but the difference between that likelihood and a 50-50 chance are negligible.) There are 128 possible sequences of boys and girls, of which 35 include four boys and 3 girls. So, the correct answer is = 35/128 = 0.27344. About 3 in 11. Here's a new question: Is this also the probability that all seven of them will be boys? Why (not)?

What is the probability that the sex order will be boy - girl - girl - girl - boy - boy - boy?

The answer depends on how you interpret the question. If one assumes that there are four boys and three girls (and this would have been known prior to the birth) then proceed as follows. There are 35 (the same number you see above--and for good reason) ways to place the four boys in four positions in the order. This particular ordering is exactly one of these 35 ways. So, the solution is 1 in 35 = 0.02857 or 2.857%.

Now, if we don't assume there will be four boys and three girls (that is, we ignore the previous question), then each child is equally likely to be either sex. There's only one way for the given sequence (BGGGBBB) to occur; there are 128 possible sequences (2⁵ = 128). The probability is then 1/128.
To briefly get technical about it. . .I'm distinguishing between two probabilities involving the birth of seven children
> The (unconditional) probability of the sex order being BGGGBBB (no information assumed about the prevalence of each gender)..
> The (conditional) probability of the sex order being BGGGBBB given that there are four boys and three girls.

Here's a new question: Is the probability that the order is girl - girl - girl - boy - boy - boy - boy (that is, "Ladies first!") the same? Why (not)?

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