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Given the following:
1. Identical twins always belong to the same gender
2. Probability of 2 people being twins is 2%
3. Probability of 2 people being identical twins is 0.16%
4. Probability of 2 people being conjoined twins is 1 in 50000
5. Gender ratio of male:female is 50:50

Some of the given information might be just for creating confusion.

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    it certainly is causing me confusion - 3. Probability of 2 people being identical twins is 0.16% - I wonder what they mean by that? Do they mean any two people, or do they mean people already know to be siblings. – 2017-02-06
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    Is it possible they mean something like "the probability that a randomly selected birth results in twins is $2\%$"? (and so on). – 2017-02-06
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    @lulu - that's what my guess would be, half of identical twins are male-male - but how often are non-identical twins male-male? – 2017-02-06
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    @Cato The probability of any two people being identical twins is 0.16% – 2017-02-06
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    @Cato I just looked it up. Twin births are about $3\%$ of births....and about one third of twin births are identical twins. I don't think there is a gender bias (but might be wrong about that). – 2017-02-06
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    I was thinking that the gender part is given just to add confusion..wouldn't gender be an independent event here? – 2017-02-06
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    @user980956 But that doesn't make sense....there are $7$ billion people in the world. the probability that any two people are even closely related is effectively $0$. I think they must mean something else... – 2017-02-06
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    I would say, that .16/2 = 8% of twins are identical, however half of twins are male female, so it could be 16% - answer = .16 – 2017-02-06
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    @lulu The numbers provided here might be realistic...they are just for the sake of the question – 2017-02-06
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    I tried to look up gender bias in twin births...I didn't get anything solid but it looks fairly unbiased (again, I might have it wrong). Of course if you have boy-girl twins then they are obviously not identical. – 2017-02-06
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    @Cato I think conditional probability needs to be taken into consideration here – 2017-02-06
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    @user980956 you mean unrealistic, I think. Sure...but I wouldn't think they'd be wildly unphysical. But of course you are right in spirit...trying to guess what the author meant is frustrating and uncertain. – 2017-02-06

2 Answers 2

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The question here is not well worded, especially as a math question. It's possible, however, to guess at what the given data means, and at what tacit assumptions apply in order to get an answer.

A way to interpret things is that, for every $10{,}000$ births, there are $200$ pairs of twins (i.e., $2\%$). Of these, $16$ pairs (i.e., $0.16\%$) are identical twins, so the $184$ other pairs are fraternal. Since identical twins must be of the same sex, there will be $8$ boy-boy twins in that subgroup. For fraternal twins, however, the sex of each twin is presumably independent, so only a quarter of that subgroup, or $46$ will be boy-boy. Thus there are $8+46=54$ boy-boy pairs of twins, of which $8$ are identical, for a probability of $8/54\approx0.148$

(I see Cato posted essentially the same answer moments before me.)

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assuming 1/4 of non-identical twins are male/male and 1/2 of identical twins are male/male

P(male-male twin) = .08 + (2 - .16) / 4 = .54

P(Identical twin|male-male twin) = P(Identical twin AND male-male twin) / P(male-male twin) = .08 / .54 = 4 / 27