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I have the following question on probability:

"Twins can either be identical or fraternal. Identical, also called monozygotic, twins form when a single feritilized egg splits into two genetically identical parts. Consequently, identical twins always have the same set of genes. Fraternal, also called dizygotic, twins develop when two separate eggs are fertilized and implant in the uterus. The genetic connection of fraternal twins is no more or less the same as siblings born at separate times. A Los Angeles country scientist wishing to know the current fraction of twin pairs born in the country that are identical twins has assigned a county statistician to study this issue. The statistician initially requested each hospital in the county to record all twin births, indicating whether the resulting twins were identical or not. The hospitals, however, told her that to determine whether newborn twins were identical was not a simple task, as it involved the permission of the twins's parents to perform complicated and expensive DNA studies that the hospitals could not afford. After some deliberation, the statistician just asked the hospitals for data listing all twin births along with an indication as to whether the twins were of the same sex. When such data indicated that approximately 64% of twin births were same-sexed, the statistician declared that approximately 28% of all twins were identical. How did she come to this conclusion?"

I think it is something to do with Bayes' formula. But, I am not able to get how 28% of all twins are identical.

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If you assume that the probability of fraternal twins being the same sex is 50%, then I agree with the logic. I'm not completely sure it might not be a bit higher than that due to other biological factors (conception timing would be similar, which might drive this higher).

In any case, if the probability of identical is p, then the probability of same sex is 100% * p + 50% * (1-p) = 1/2 + 1/2 p

Solving 0.64 = 0.50 + 0.5p, p = 0.28.

Again, I suspect (from a biological / mechanical perspective) that non identical twins are the same sex at higher than 50%, thus the probability of identical twins would be less than 28%.

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