Probability of two women giving birth on their birthday A person's Facebook account has 435 connections.
Of those connections 3 have given birth to daughters in Feburary.
Of those three births two have occurred on the mother's birthday.
Question: What is probability of two women giving birth to girls on their birthday?
Here's my understanding:
Assumes probability of boy/girl each at 0.5
You have two independent events. 
Event #1:Mother #1 giving birth to female baby on Mother #1 birthday.
Event #2:Mother #2 giving birth to female baby on Mother #2 birthday.
So what is the probability of event #1?
It is the probability of choosing two people Mother #1 and Baby Female #1  at random on same birthday.
The probability of Mother #1 birthday any specific day of the year is 1/365
The probability of Female Baby  #1 birthday  any one specific day of the year is 1/365 * 1/2
Event #1 = Probability of Both A1 and B1 = 
             P(A1) * P(B1) = 
             1/365 * 1/365 * 1/2 = 
             1/266,450
So what is the probability of event #2?
The same as event #1.
What is the probability of Event #1 AND Event #2
Probability = P(event #1) * P(Event #2) = 
               1/266,450 * 1/266,450= 
               1/70,995,602,500
Approximately 1 in 71 Billion
It this an accurate explanation?
Would the probability change if:


*

*Only 3 of the 435 connections were female?

*All 435 connections were female?
 A: There are two main sources of fuzziness in your question. The first is how many of the 435 contacts are females; some people have a very significant skew in the gender of their contacts. The second is how many children (and in particular daughters) those females have given birth to; the answer if obviously quite different if we are looking at the contacts of a $15-$year old, or at the contacts of a $51-$year old.
In general, a mother of $d$ daughters has probability $\approx$ $(\frac{364}{365})^d$ of having none of them born on her birthday. The $\approx$ is due to some years having $366$ days (which increases the probability) and to the fact that children are not born evenly throughout the year (though close enough). For smallish $d$, we can then approximate the probability of a mother of $d$ daughters having at least one born on her own birthday as $1-(\frac{364}{365})^d\approx \frac{d}{365}$.
If we have $m$ such mothers, the probability that exactly $s$ have the same birthday as a daughter is then $\approx{m\choose s} (\frac{d}{365})^s (1-\frac{d}{365})^{m-s}$. For smallish $s$ and $d$, and for $md$ significantly smaller than $365$, we can approximate this further as $\approx\frac{1}{s!}(\frac{md}{365})^s$.
So, if all the contacts are mothers (i.e. $m=435$), and each has $1$ daughter ($d=1$), for $s=2$ we have a probability $\approx{435\choose 2} (\frac{1}{365})^2 (1-\frac{1}{365})^{435-2}$ or a little over $20\%$ that exactly $2$ of them will have a daughter with the same birthday. Not that unlikely! Note that $md > 365$ in this case, so we can't quite use the most simplified formula. On the other hand, if we only have $3$ mothers ($m=3$), we can use the most simplified formula and compute the probability as $\approx \frac{1}{2!}(\frac{3\cdot 1}{365})^2$ or about $1$ in $30000$ - far less likely, but not quite impossible.
