Identical Twins
Once, I gave a puzzle to my students as homework just to check their level of attention.
Puzzle. A family has two identical twins. One of them is a boy, what is the probability that the other one is a boy?
What can I say? Some of the students didn’t pay attention and gave weird answers like 1/2, 1/3, and 2/3.
The twins are IDENTICAL. The other one has to be a boy! Duh!
One of the students was well-educated and mentioned that it is theoretically possible for different twins to be different genders, though this is extremely rare. When one fertilized egg splits into two, producing two embryos, the genetic material of both eggs is the same, almost. Some errors during splitting are possible, and it seems that some very particular errors can lead to the identical twins being identified as a boy and a girl. I never knew that before!
However, one student thought outside the box. In his vision, a family adopted two identical twins who aren’t each other’s twins, just happen to be identical twins with someone else.
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Izzy Grosof:
Another possibility is that one of the twins is trans, while the other is cis.
28 December 2024, 12:56 pmKicab Castaneda-Mendez:
Whether identical twins means they are of the same gender or scientifically they can be of different gender, the probability the other twin is a boy is either 0 or 1. The fact is the other child exists and surely the parents know what gender the child is. For them, the probability is 0 if different gender and 1 if the same.
This is the same as in Gardner’s Two-Children Problem. Consider one version of the Two-Children Problem: “Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?” Tanya wrote in her article “Martin Gardner’s Mistake,” that “There was never any controversy about Mr. Jones who with probability 1/2 has two girls.”
There is a controversy. Surely, Mr. Smith knows whether he has two girls or not. Mr. Smith’s probability is 1 in the former case and 0 in the latter. That probability of 0 or 1 is based on the fact. The 1/2, 1/3, 2/3 and other probabilities not 0 or 1, are not. They assume either the child does not exist yet or that the two-child family will be selected randomly. Now that the family has been selected the probability is 0 or 1.
Reword the problem #1: Mr. Jones has two daughters. You only know the older child is a girl. What is the probability that both children are girls? Clearly, the probability is 1. Reword again #2: Mr. Jones has a son and daughter. You only know the older child is a girl. What is the probability that both children are girls? Clearly, the probability is 0. Since only one of these is the case, then the probability is either 0 or 1–even if you don’t know which.
Once an event occurs, the original sample space is split into subspaces where only one is the case. For #1, the original sample space {(girl, girl), (girl, boy)} splits into {(girl, girl)} and {(girl, boy)}, yielding probabilities of 1 and 0, respectively.
11 January 2025, 10:48 amKicab Castaneda-Mendez:
Forgot the reference:
Kicab Castaneda-Mendez,
Mine Çetinkaya-Rundel (Column Editor) & Maria Tackett (Column Editor) (2024)
Taking a Chance in the Classroom: Puzzling Probabilities of
Probability Puzzles, CHANCE, 37:3, 60-65, DOI: 10.1080/09332480.2024.2415844 https://doi.org/10.1080/09332480.2024.2415844
Tanya, let me know if you canmot access the article.
12 January 2025, 1:59 pm