I think I might have partially asked this already in the comments on somebody else's question. As far as I remember it was in an anthology of four or five novella's or "long" short stories....I read it sometime in the 1970's but it might have been written earlier.
A woman is standing naked at a municipal swimming pool, no inhibition on nudity by any other users (everyone is nude) in this future society, she is actually flaunting her pregnancy and the fact she won the lottery to bear a child. The population is controlled by this measure and is referred to as winning a "Bingo Baby"
There's a few chapters on her life and marriage but then it is noticed a lot of other women are also pregnant but unlicensed, there is a race on to find what's going on, eventually an agent realizes that all the women have used the municipal swimming pools recently.
They track down clues about a rogue scientist who has been modifying his sperm to super-survive, he's been releasing it in pools all over North America.
That's all I remember, maybe then a manhunt but I'm not sure
Edit: I read this before I'd read any of Niven's Known Space books and I remember noting the similarity (about a lottery that allowed child bearing) when I finally got into Ringworld.