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2nd problem is 1 out of 1000 births result in fraternal twins; 1 out of 1500 births result in identical twins. Identical twins must be the same sex while the sexes of fraternal twins are independent. If two girls are twins what is the probability they are fraternal twins?
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This is a good example for conditional probabilities (See
http://mathworld.wolfram.com/Conditi...bability.html), but you can tackle the problem intuitively.
It helps to bring the probabilities to a common denominator. The smallest common denominator of 1000 and 1500 is 3000, but for reasons that will be clear in a moment, I want the numerator for the probability of fraternal twins to be an even number, so I'll choose 6000 as the common denominator.
Hence, P(fraternal twins) = 6/6000 and P(identical twins) = 4/6000.
Since the genders of fraternal twins are independent, half the fraternal twin births will be same-sex, the other half will be mixed-sex. This gives me
P(mixed-sex fraternal twins) = 3/6000 and P(same-sex fraternal twins) = 3/6000.
Now let's get back to the problem: Given that a same-sex twin birth has occurred, what is the probability that it is a fraternal twin birth? To answer this, we simply have to find the ratio of fraternal same-sex twin births to the total number of same-sex twin births.
Out of 6000 births, there are 3 fraternal same-sex twin births and 4 identical (necessarily same-sex) twin births, for a total of 7 same-sex twin births. Of those 7, 3 are fraternal, so the answer is 3/7.
I'll leave it as an exercise for the reader to convince themselves that any other choice of population size would yield the same result.
Hope this helps,
Carsten.