Probability of 4 parents sharing 2 birthdays I'm a math idiot and I'm trying to figure something out. My mother was born on April 10th as was my friend's mother. My father was born on July 4th as was my friend's father.
How do I calculate the probability of that happening?
EDIT: How would one calculate the probability of 2 pairs of shared birthdays? I think this is the answer I'm looking for
Halp.
-Brett

Since you don't care about overlaps (you said "I'm looking for the probability of all 4 falling on those exact dates."), you take the probability your mom's birthday is April 10th, and multiply that by the probability your friend's mom's birthday is April 10th, then you multiply that by the probability your dad was born on July 4th, and multiply that by the probability your friend's dad was born on July 4th.
This means $\frac{1}{365}*\frac{1}{365}*\frac{1}{365}*\frac{1}{365} = (\frac{1}{365})^4 = 5.63*10^{-11}$ very slim (THIS IS EXCLUDING LEAP YEARS). Multiply by 100 to get a percent: $5.63*10^{-9}$%
Bear in mind, that since you said you only care about birthdays falling on specific dates, the probability your mom's b-days are April 10th, and dad's are July 4th, is THE EXACT SAME PROBABILITY of your mom's birthday being January 28th, your friend's mom's Decmber 2nd, your Dad's April 11th, and your friend's dad's being October 30th.
If you only care about 2 pairs of the same birthday, the probability would actually be higher than for any select birthday assigned (the probability above). The probability of 2 pairs of shared birthdays gets more complicated as well...
for that, you can now see @CoveredInChocolate's answer
Fun fact: if there's a group of 23 people, there's over a 50% chance that 2 or more of them share a birthday.
