a weather girl predicts there is a 50% chance of rain and a 10% chance that the rain will turn into snow. What is the probability that it will snow?

Agree, you have a "dependent" condition, so
prob = (.5)(.1) = .05 or 5%

To answer your other question:

In the USA, and perhaps other English speaking countries, most people get their weather forecasts by watching television, where the presenter is usually a young woman, especially in the morning and afternoon hours of broadcasting. As these women are usually selected for attractiveness and speaking ability as well as meteorological knowledge, often in that order, they are often referred to as "weather girls", even though this term might appear rather disrespectful and male-chauvinistic.

The female cheerleaders of the LA Lakers baskeball team, who are called the "Laker Girls", and who resemble many of the weather "girls" except in their attire, probably do not resent the name.

I think a "weather girl" is a member of a radical left, originally student, group once associated with the SDS (Students for a Democratic Society). They were a particularly militant branch of students opposed to the Vietnam War under the leadership of Bill Ayres.