Four people share a birthday

I was born on my mother’s 21st birthday – 12th February 1968. A few years ago I was chatting with the elderly gentleman who lived opposite us (he has since died), and I mentioned that it was my birthday next week. “Oh really,” he said. “Which day?” “On Wednesday the 12th.” “Oh, that’s my birthday too,” he said. “Really?” I said. “What a coincidence! It’s always been a special day in our family, because it’s my Mum’s birthday as well.” “Ah yes,” he said. “Mine too.” He was quite old and a little deaf, so I thought maybe he’d misunderstood me. “No, you see I was born on my mother’s birthday.” “Yes,” he said patiently, “me too!” I had never (before or since) met another person who was born on their mother’s birthday, although I’m sure there must be quite a few around. But what are the chances of meeting someone in the same situation, whose birthday was also 12th February?