Royal babies math question

Apparently mother of two had her two kids born the exact same days as Princess Kate. A reporter just asked me what the probability is of that. As long as I went through the trouble, let me present my rough answer below. Let's look at the UK as our eligible universe.

1. Assuming a nine-month pregnancy, the time between the birth of Prince George and the conception of Princess Charlotte is 379 days.
2. Let's assume that only women ages 20-40 are eligible.
3. The population of the UK in that age range is 16,924,000 (source: wikipedia)
4. Let's say half of them are women, so 8,462,000 eligible mothers.
5. The fertility rate in the UK is 1.92. Using the Poisson distribution, the probability of two or more children is 69.83%. Note that the woman in question may have more kids.
6. The number of women of children bearing age who will have two or more children in the UK is 8,462,000 * 69.83% = 5,909,169
7. Let's say that of these eligible mothers, the first child (if any) could be born at random according to a uniform distribution between ages 20 and 37.
8. The number of women in the UK giving birth to her first child, who will eventually have two or more children, is 5,909,169/(365*17), per day = 952.32
9. Let's assume the number of days between the first of the first child and conception of the second is distribution via the exponential distribution with a mean of two years, or 730 days. I admit the two years is rather arbitrary, but I have to assume something. So, the probability of this exact age difference, to the day, is EXP(-(1/730)*825)-EXP(-(1/730)*826) = 0.000442148
10. The expected number of women in the UK to become a mother for the second time, with an age difference equal to Princess Kate to the day, is 952.32 * 0.000429714 = 0.421068.
11. The probability this happened more than once in the UK = 1-exp(-0.421068) = 34.37%.

By the way, the probability this happened in the US is 86.32%.

The question for the poll is do you think my probability is too low, too high, or about right?

Reminds me of one of my favorite questions:

What is the probability that something improbable will happen?

I voted Wiz is probably close. Also, I wonder if women who had a match on the first one and had a reasonable chance of matching the second one could use some sort of trickery to make sure there was a second match and possibly benefit in some weird way