Yes, it's possible for a mutation in the chromosome to produce otherwise-identical twins with opposite genders. But generally no, identical twins have the same gender. (but fraternal twins have different genders just as often as any siblings)
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If they were not then they would not be called identical twins!
Identical twins are formed from a single egg that splits into two. Fraternal twins are formed from two separate eggs that are both separately fertilized. So identical twins have identical DNA, and fraternal twins do not.
Twins are not necessarily identical. They can be fraternal as well. Identical twins result when one fertilized egg splits in two. Fraternal twins result from two fertilized eggs.
They are twins!
There is no simple answer to this question because the genders of children depend on the parents genes and age and so are not independent events. Also, the probabilities of the two genders are not equal. Further, some of the children may be identical twins or triplets. Assuming, however, that the genders are independent, the answer is 0.3872, approx.