Wednesday, November 25, 2009

Turnabout Time: Ana Has Questions For Us!

1. Ana is fascinated by languages, which is no surprise. She has a question for us:

"I can say 'I will bring a dessert to you,' and I can say 'I will take a dessert to you.' What is the difference between 'bring' and 'take?'"

2. Ana is intrigued by the famous "birthday problem:" how many people must you have gathered together to have a fifty per cent chance that two of them will have the same birthday?

You would think, since there are 365 days in a year, that you would need quite a crowd of people, but not so! You only need 23 people! We have tested this many times over the years with classes of students, usually averaging about 25 individuals, and we have found a match at least half the time. The explanation may be found here.

But Ana has a different question: suppose you have 25 people in a room and no matches. One student walks in late. What are the chances that he or she will have the same birthday as someone already present?

She reasons this way: with 25 people and no match, that leaves 365 - 25 empty days, or 340 days. So when that person shows up, will his or her birthday be one of the 25, or one of the 340? She thinks the odds are 25 to 340 against, or 13.6 to one. Any mathematician will tell you that is correct. Yet the birthday problem suggests that the real answer should be approximately 50/50.

Ana knows that obviously something is wrong with her reasoning. She understands the math in both situations, but not where her thinking is in error. Can anyone explain this to her?

(Explanations here.)


Danielle said...

Once upon a time, I might have understood that math problem, been able to work it out and explain it to someone. Now I look at it and go, huh? That's exercising brain matter that's been asleep for some time!

Al said...

Alas, there was never a time when I would have understood that problem. Still, ever the industrious researcher, I sought out and managed to find not only a top-flight mathematician but also a gifted teacher of mathematics (in one person, a most rare serendipity), who understood what I did NOT understand, and explained it so I could: they are two entirely different situations. My sincere thanks to that Mystery Mathematician, whose clarification is given in the answer (in the link above).