Comment and Answers

Hi class!


Reading through your responses I am quite impressed. Many of you took different approaches to solving the questions as well as debated with each other how they went about coming up with the answers. Well done! Here are the answers to the problems. Again, I am not marking you based on if your answer is correct but how you went about coming up with your answer.
____________________________________________

Problem 1 Solution

The position in line that gives you the best chance of being the first duplicate birthday is 20th.


Problem 2 Solution

We assume that each birth is an independent event, for which the probability of a boy is the same as the probability of a girl. There are, then, three possibilities for your colleague's family, all equally likely:

•Boy, Boy, Girl

•Boy, Girl, Boy

•Boy, Girl, Girl

Therefore there is a 2/3 chance that the colleague has two boys and a girl, and a 1/3 chance he has two girls and a boy.

Problem 1: The Birthday Line

At a movie theater, the manager announces that a free ticket will be given to the first person in line whose birthday is the same as someone in line who has already bought a ticket.  You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, and that birthdays are uniformly distributed throughout a 365 day year, what position in line gives you the best chance of being the first duplicate birthday?

Problem 2: Three Children

On the first day of a new job, a colleague invites you around for a barbecue.  As the two of you arrive at his home, a young boy throws open the door to welcome his father.  “My other two kids will be home soon!” remarks your colleague.


Waiting in the kitchen while your colleague gets some drinks from the basement, you notice a letter from the principal of the local school tacked to the noticeboard.  “Dear Parents,” it begins, “This is the time of year when I write to all parents, such as yourselves, who have a girl or girls in the school, asking you to volunteer your time to help the girls' soccer team.”  “Hmmm,” you think to yourself, “clearly they have at least one of each!”


This, of course, leaves two possibilities: two boys and a girl, or two girls and a boy.  Are these two possibilities equally likely, or is one more likely than the other?