Some countries has prohibited smoking in public areas in buildings. Use the concept of private and social costs and benefits to explain this policy.

Various countries and states have adopted this policy for a variety of reasons, so working through a particular private and social cost-benefit analysis isn't necessarily going to apply to each situation.
This question also has a fairly large, unnamed variable present in the question: what is the entity being used for the private cost-benefit analysis? It could be an actual person who either is or isn't a smoker. It could be the business that smokers are no longer allowed smoking in, or it could be the entire state/country.
I think it is most appropriate to examine a particular business for this question. Let's use a restaurant for our private and social cost-benefit analysis. The restaurant's private costs are not likely to change because of the no smoking policy. The restaurant still has to pay for things like rent, wages, utilities, and the food and drink being prepared and served. Whether a customer can or can't smoke in the restaurant doesn't change those private costs.
A private benefit of eliminating smoking in the restaurant could be a reduction in cleaning. The restaurant saves money by not having to supply ash trays. The restaurant also saves money by not having to purchase and operate fan systems designed to pull smoke from the room.
A possible social benefit comes from an overall increase in air quality. Employees are no longer breathing in all that second hand smoke, so the staff is all-around healthier. This could actually contribute to a private benefit in that the business is saving on healthcare-related costs due to the fact that employees are working in a healthier environment.
https://www.economicshelp.org/blog/glossary/social-benefit/

https://www.economicshelp.org/blog/glossary/social-cost/