Suppose Mary is a marketer for Greyhound, a low-cost bus service. She is considering a special promotion that will give away free snacks to everyone on the bus if at two or more people on a trip have the same birthday.

Reviewing travel logs, Mary calculates an average of about 40 people per bus trip. The cost for snacks would be too high if more than 20% of the bus trips would require providing snacks. 

This seems like a pretty good promotional idea to Mary at first. With 365 days in a year, she reasons that 40/365 ≈ 11%, well below the 20% threshold. However, much to her surprise, she randomly checks 10 bus trips (each with 40 people) and a shocking 9 of them had at least one pair of people with the same birthday! As exciting as the promotion might be, the company simply can't afford snacks on that many of the trips.

For your initial post, respond to the following questions. Grace comes to you for help. You will need to post before you can see anyone else's reply.

• Do you agree or disagree with Grace's estimate of 11%? Why or why not? 
• Was she simply unlucky in checking the bus trips? Why or why not?
• support your response with mathematical reasoning.
• What would happen if there were 50 people on the bus? 60 people? 30?